Linear Regression Final Project

df <- as_tibble(read.table("winequality-red.csv", sep=";", header = TRUE))
df_white <- as_tibble(read.table("winequality-white.csv", sep=";", header = TRUE))

index = sample(1:nrow(df), 0.7*nrow(df)) 

train = df[index,] # Create the training data 
test = df[-index,] # Create the test data

dim(train)

## [1] 1119   12

dim(test)

## [1] 480  12

Rudimentary LM on whole dataset

lm_basic <- lm(data = df, quality ~ .)

summary(lm_basic)

## 
## Call:
## lm(formula = quality ~ ., data = df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.68911 -0.36652 -0.04699  0.45202  2.02498 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           2.197e+01  2.119e+01   1.036   0.3002    
## fixed.acidity         2.499e-02  2.595e-02   0.963   0.3357    
## volatile.acidity     -1.084e+00  1.211e-01  -8.948  < 2e-16 ***
## citric.acid          -1.826e-01  1.472e-01  -1.240   0.2150    
## residual.sugar        1.633e-02  1.500e-02   1.089   0.2765    
## chlorides            -1.874e+00  4.193e-01  -4.470 8.37e-06 ***
## free.sulfur.dioxide   4.361e-03  2.171e-03   2.009   0.0447 *  
## total.sulfur.dioxide -3.265e-03  7.287e-04  -4.480 8.00e-06 ***
## density              -1.788e+01  2.163e+01  -0.827   0.4086    
## pH                   -4.137e-01  1.916e-01  -2.159   0.0310 *  
## sulphates             9.163e-01  1.143e-01   8.014 2.13e-15 ***
## alcohol               2.762e-01  2.648e-02  10.429  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.648 on 1587 degrees of freedom
## Multiple R-squared:  0.3606, Adjusted R-squared:  0.3561 
## F-statistic: 81.35 on 11 and 1587 DF,  p-value: < 2.2e-16

Volatile acidity, chlorides, total sulfur dioxide, sulphates and alcohol are all very statistically significant, although our adjusted \(R^2\) is still quite low.

Ridge regression

train_control <- trainControl(method  = "cv", number = 5)

model_ridge <- train(quality ~ .,
                     data = df,
                     method = "ridge",           # method
                     trControl = train_control)        # cross validation

model_ridge

## Ridge Regression 
## 
## 1599 samples
##   11 predictor
## 
## No pre-processing
## Resampling: Cross-Validated (5 fold) 
## Summary of sample sizes: 1281, 1280, 1279, 1278, 1278 
## Resampling results across tuning parameters:
## 
##   lambda  RMSE       Rsquared   MAE      
##   0e+00   0.6506816  0.3540442  0.5043230
##   1e-04   0.6506802  0.3540472  0.5043209
##   1e-01   0.6504990  0.3548844  0.5033013
## 
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was lambda = 0.1.

#something doesn't seem to be working here

model_stepwise <- train(quality ~ .,
                        data = df,
                        method = "glmStepAIC",
                        trControl = train_control)

## Start:  AIC=2569.31
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - residual.sugar        1   547.09 2567.5
## - density               1   547.15 2567.6
## - fixed.acidity         1   547.20 2567.7
## - citric.acid           1   547.72 2569.0
## <none>                      547.01 2569.3
## - free.sulfur.dioxide   1   548.19 2570.1
## - pH                    1   549.84 2573.9
## - total.sulfur.dioxide  1   553.62 2582.7
## - chlorides             1   553.94 2583.4
## - sulphates             1   565.29 2609.3
## - volatile.acidity      1   574.97 2631.1
## - alcohol               1   584.22 2651.5
## 
## Step:  AIC=2567.49
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + chlorides + 
##     free.sulfur.dioxide + total.sulfur.dioxide + density + pH + 
##     sulphates + alcohol
## 
##                        Df Deviance    AIC
## - density               1   547.15 2565.6
## - fixed.acidity         1   547.21 2565.8
## - citric.acid           1   547.79 2567.1
## <none>                      547.09 2567.5
## - free.sulfur.dioxide   1   548.33 2568.4
## - pH                    1   550.52 2573.5
## - total.sulfur.dioxide  1   553.64 2580.7
## - chlorides             1   554.03 2581.6
## - sulphates             1   565.51 2607.8
## - volatile.acidity      1   575.06 2629.3
## - alcohol               1   599.28 2682.0
## 
## Step:  AIC=2565.64
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + chlorides + 
##     free.sulfur.dioxide + total.sulfur.dioxide + pH + sulphates + 
##     alcohol
## 
##                        Df Deviance    AIC
## - fixed.acidity         1   547.21 2563.8
## - citric.acid           1   547.87 2565.3
## <none>                      547.15 2565.6
## - free.sulfur.dioxide   1   548.41 2566.6
## - pH                    1   552.41 2575.9
## - total.sulfur.dioxide  1   554.01 2579.6
## - chlorides             1   554.30 2580.2
## - sulphates             1   565.85 2606.6
## - volatile.acidity      1   576.26 2629.9
## - alcohol               1   639.98 2764.1
## 
## Step:  AIC=2563.77
## .outcome ~ volatile.acidity + citric.acid + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - citric.acid           1   547.96 2563.5
## <none>                      547.21 2563.8
## - free.sulfur.dioxide   1   548.50 2564.8
## - total.sulfur.dioxide  1   555.00 2579.9
## - chlorides             1   555.28 2580.5
## - pH                    1   555.47 2580.9
## - sulphates             1   566.06 2605.1
## - volatile.acidity      1   577.87 2631.5
## - alcohol               1   640.60 2763.3
## 
## Step:  AIC=2563.52
## .outcome ~ volatile.acidity + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      547.96 2563.5
## - free.sulfur.dioxide   1   549.56 2565.3
## - pH                    1   555.83 2579.8
## - total.sulfur.dioxide  1   556.73 2581.8
## - chlorides             1   557.02 2582.5
## - sulphates             1   566.40 2603.9
## - volatile.acidity      1   582.31 2639.3
## - alcohol               1   641.95 2764.0
## Start:  AIC=2560.19
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - fixed.acidity         1   542.71 2558.2
## - density               1   542.72 2558.2
## - citric.acid           1   542.73 2558.2
## - residual.sugar        1   542.73 2558.2
## <none>                      542.71 2560.2
## - free.sulfur.dioxide   1   544.78 2563.1
## - pH                    1   545.45 2564.6
## - total.sulfur.dioxide  1   551.59 2579.0
## - chlorides             1   554.08 2584.7
## - sulphates             1   563.03 2605.2
## - volatile.acidity      1   570.15 2621.3
## - alcohol               1   581.41 2646.3
## 
## Step:  AIC=2558.2
## .outcome ~ volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - density               1   542.73 2556.2
## - citric.acid           1   542.73 2556.2
## - residual.sugar        1   542.74 2556.3
## <none>                      542.71 2558.2
## - free.sulfur.dioxide   1   544.79 2561.1
## - pH                    1   548.08 2568.8
## - total.sulfur.dioxide  1   552.05 2578.0
## - chlorides             1   554.64 2584.0
## - sulphates             1   563.53 2604.4
## - volatile.acidity      1   570.25 2619.6
## - alcohol               1   596.88 2678.0
## 
## Step:  AIC=2556.23
## .outcome ~ volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - citric.acid           1   542.74 2554.3
## - residual.sugar        1   542.78 2554.4
## <none>                      542.73 2556.2
## - free.sulfur.dioxide   1   544.81 2559.1
## - pH                    1   548.08 2566.8
## - total.sulfur.dioxide  1   552.36 2576.8
## - chlorides             1   554.93 2582.7
## - sulphates             1   564.71 2605.1
## - volatile.acidity      1   572.07 2621.6
## - alcohol               1   629.63 2744.3
## 
## Step:  AIC=2554.26
## .outcome ~ volatile.acidity + residual.sugar + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - residual.sugar        1   542.79 2552.4
## <none>                      542.74 2554.3
## - free.sulfur.dioxide   1   544.92 2557.4
## - pH                    1   549.19 2567.4
## - total.sulfur.dioxide  1   552.66 2575.4
## - chlorides             1   555.26 2581.4
## - sulphates             1   564.85 2603.4
## - volatile.acidity      1   580.85 2639.1
## - alcohol               1   632.02 2747.2
## 
## Step:  AIC=2552.37
## .outcome ~ volatile.acidity + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      542.79 2552.4
## - free.sulfur.dioxide   1   545.06 2555.7
## - pH                    1   549.40 2565.9
## - total.sulfur.dioxide  1   552.68 2573.5
## - chlorides             1   555.26 2579.4
## - sulphates             1   564.85 2601.4
## - volatile.acidity      1   580.86 2637.1
## - alcohol               1   633.89 2749.0
## Start:  AIC=2514.74
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - free.sulfur.dioxide   1   524.58 2513.8
## - pH                    1   524.90 2514.5
## <none>                      524.16 2514.7
## - density               1   525.05 2514.9
## - residual.sugar        1   525.54 2516.1
## - fixed.acidity         1   525.55 2516.1
## - citric.acid           1   526.83 2519.2
## - chlorides             1   528.06 2522.2
## - total.sulfur.dioxide  1   528.46 2523.2
## - sulphates             1   547.27 2567.9
## - volatile.acidity      1   556.39 2589.0
## - alcohol               1   561.00 2599.6
## 
## Step:  AIC=2513.75
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + total.sulfur.dioxide + density + pH + sulphates + 
##     alcohol
## 
##                        Df Deviance    AIC
## - pH                    1   525.19 2513.2
## <none>                      524.58 2513.8
## - density               1   525.58 2514.2
## - fixed.acidity         1   526.14 2515.6
## - residual.sugar        1   526.17 2515.6
## - citric.acid           1   527.69 2519.3
## - chlorides             1   528.34 2520.9
## - total.sulfur.dioxide  1   529.35 2523.3
## - sulphates             1   548.25 2568.2
## - volatile.acidity      1   558.92 2592.8
## - alcohol               1   561.30 2598.3
## 
## Step:  AIC=2513.24
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + total.sulfur.dioxide + density + sulphates + 
##     alcohol
## 
##                        Df Deviance    AIC
## <none>                      525.19 2513.2
## - residual.sugar        1   527.76 2517.5
## - citric.acid           1   528.10 2518.3
## - density               1   528.28 2518.8
## - chlorides             1   528.48 2519.2
## - total.sulfur.dioxide  1   529.58 2521.9
## - fixed.acidity         1   531.86 2527.4
## - sulphates             1   549.97 2570.2
## - volatile.acidity      1   559.48 2592.1
## - alcohol               1   568.64 2612.9
## Start:  AIC=2510.97
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - citric.acid           1   522.65 2509.0
## - density               1   523.03 2510.0
## - residual.sugar        1   523.10 2510.1
## - fixed.acidity         1   523.15 2510.3
## - pH                    1   523.40 2510.9
## <none>                      522.62 2511.0
## - free.sulfur.dioxide   1   525.44 2515.9
## - chlorides             1   526.74 2519.0
## - total.sulfur.dioxide  1   529.60 2525.9
## - volatile.acidity      1   540.25 2551.4
## - sulphates             1   547.99 2569.6
## - alcohol               1   556.17 2588.6
## 
## Step:  AIC=2509.03
## .outcome ~ fixed.acidity + volatile.acidity + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - density               1   523.05 2508.0
## - residual.sugar        1   523.11 2508.2
## - fixed.acidity         1   523.16 2508.3
## - pH                    1   523.42 2508.9
## <none>                      522.65 2509.0
## - free.sulfur.dioxide   1   525.62 2514.3
## - chlorides             1   527.20 2518.1
## - total.sulfur.dioxide  1   530.38 2525.8
## - volatile.acidity      1   546.62 2564.4
## - sulphates             1   548.00 2567.6
## - alcohol               1   556.47 2587.2
## 
## Step:  AIC=2508.02
## .outcome ~ fixed.acidity + volatile.acidity + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - fixed.acidity         1   523.16 2506.3
## - residual.sugar        1   523.19 2506.4
## <none>                      523.05 2508.0
## - pH                    1   525.39 2511.7
## - free.sulfur.dioxide   1   526.23 2513.8
## - chlorides             1   527.88 2517.8
## - total.sulfur.dioxide  1   531.02 2525.4
## - sulphates             1   548.44 2566.6
## - volatile.acidity      1   548.92 2567.8
## - alcohol               1   621.88 2727.4
## 
## Step:  AIC=2506.3
## .outcome ~ volatile.acidity + residual.sugar + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - residual.sugar        1   523.33 2504.7
## <none>                      523.16 2506.3
## - free.sulfur.dioxide   1   526.33 2512.0
## - chlorides             1   528.29 2516.8
## - pH                    1   528.44 2517.1
## - total.sulfur.dioxide  1   531.73 2525.1
## - sulphates             1   549.40 2566.9
## - volatile.acidity      1   549.61 2567.4
## - alcohol               1   621.92 2725.4
## 
## Step:  AIC=2504.71
## .outcome ~ volatile.acidity + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      523.33 2504.7
## - free.sulfur.dioxide   1   526.65 2510.8
## - chlorides             1   528.35 2514.9
## - pH                    1   528.85 2516.1
## - total.sulfur.dioxide  1   531.73 2523.1
## - sulphates             1   549.40 2564.9
## - volatile.acidity      1   549.76 2565.7
## - alcohol               1   624.71 2729.2
## Start:  AIC=2511.04
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - density               1   522.89 2509.6
## - fixed.acidity         1   522.89 2509.6
## - residual.sugar        1   523.37 2510.8
## - citric.acid           1   523.43 2511.0
## <none>                      522.65 2511.0
## - free.sulfur.dioxide   1   523.56 2511.3
## - pH                    1   524.16 2512.7
## - total.sulfur.dioxide  1   529.81 2526.4
## - chlorides             1   530.59 2528.3
## - sulphates             1   543.90 2560.0
## - volatile.acidity      1   552.85 2580.9
## - alcohol               1   559.46 2596.1
## 
## Step:  AIC=2509.62
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - fixed.acidity         1   522.92 2507.7
## - residual.sugar        1   523.37 2508.8
## - citric.acid           1   523.66 2509.5
## <none>                      522.89 2509.6
## - free.sulfur.dioxide   1   523.89 2510.1
## - pH                    1   526.30 2515.9
## - total.sulfur.dioxide  1   530.34 2525.7
## - chlorides             1   531.21 2527.8
## - sulphates             1   544.30 2559.0
## - volatile.acidity      1   553.99 2581.5
## - alcohol               1   624.48 2734.7
## 
## Step:  AIC=2507.69
## .outcome ~ volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - residual.sugar        1   523.42 2506.9
## <none>                      522.92 2507.7
## - citric.acid           1   523.79 2507.8
## - free.sulfur.dioxide   1   523.95 2508.2
## - pH                    1   528.12 2518.4
## - total.sulfur.dioxide  1   531.23 2525.9
## - chlorides             1   532.21 2528.2
## - sulphates             1   544.54 2557.5
## - volatile.acidity      1   555.45 2582.9
## - alcohol               1   625.67 2735.2
## 
## Step:  AIC=2506.93
## .outcome ~ volatile.acidity + citric.acid + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - citric.acid           1   524.13 2506.7
## <none>                      523.42 2506.9
## - free.sulfur.dioxide   1   524.64 2507.9
## - pH                    1   528.72 2517.8
## - total.sulfur.dioxide  1   531.48 2524.5
## - chlorides             1   532.56 2527.1
## - sulphates             1   544.69 2555.9
## - volatile.acidity      1   555.53 2581.1
## - alcohol               1   627.55 2737.0
## 
## Step:  AIC=2506.66
## .outcome ~ volatile.acidity + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      524.13 2506.7
## - free.sulfur.dioxide   1   525.66 2508.4
## - pH                    1   528.78 2515.9
## - total.sulfur.dioxide  1   533.22 2526.7
## - chlorides             1   534.50 2529.7
## - sulphates             1   545.09 2554.8
## - volatile.acidity      1   560.18 2589.7
## - alcohol               1   628.94 2737.8
## Start:  AIC=3164.28
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - density               1   666.70 3163.0
## - fixed.acidity         1   666.80 3163.2
## - residual.sugar        1   666.91 3163.5
## - citric.acid           1   667.06 3163.8
## <none>                      666.41 3164.3
## - free.sulfur.dioxide   1   668.10 3166.3
## - pH                    1   668.37 3167.0
## - chlorides             1   674.80 3182.3
## - total.sulfur.dioxide  1   674.84 3182.4
## - sulphates             1   693.38 3225.7
## - volatile.acidity      1   700.03 3241.0
## - alcohol               1   712.08 3268.3
## 
## Step:  AIC=3162.96
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - fixed.acidity         1   666.81 3161.2
## - residual.sugar        1   666.93 3161.5
## - citric.acid           1   667.35 3162.5
## <none>                      666.70 3163.0
## - free.sulfur.dioxide   1   668.53 3165.3
## - pH                    1   671.02 3171.3
## - total.sulfur.dioxide  1   675.43 3181.8
## - chlorides             1   675.46 3181.8
## - sulphates             1   693.98 3225.1
## - volatile.acidity      1   701.70 3242.8
## - alcohol               1   786.37 3424.9
## 
## Step:  AIC=3161.22
## .outcome ~ volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - residual.sugar        1   667.06 3159.8
## - citric.acid           1   667.37 3160.6
## <none>                      666.81 3161.2
## - free.sulfur.dioxide   1   668.71 3163.8
## - pH                    1   673.87 3176.1
## - chlorides             1   676.75 3182.9
## - total.sulfur.dioxide  1   676.84 3183.1
## - sulphates             1   694.48 3224.2
## - volatile.acidity      1   703.04 3243.8
## - alcohol               1   787.44 3425.1
## 
## Step:  AIC=3159.84
## .outcome ~ volatile.acidity + citric.acid + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - citric.acid           1   667.54 3159.0
## <none>                      667.06 3159.8
## - free.sulfur.dioxide   1   669.13 3162.8
## - pH                    1   674.20 3174.9
## - total.sulfur.dioxide  1   676.89 3181.2
## - chlorides             1   676.89 3181.2
## - sulphates             1   694.51 3222.3
## - volatile.acidity      1   703.04 3241.8
## - alcohol               1   789.73 3427.8
## 
## Step:  AIC=3158.98
## .outcome ~ volatile.acidity + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      667.54 3159.0
## - free.sulfur.dioxide   1   669.93 3162.7
## - pH                    1   674.61 3173.8
## - total.sulfur.dioxide  1   678.32 3182.6
## - chlorides             1   678.35 3182.7
## - sulphates             1   694.60 3220.5
## - volatile.acidity      1   709.85 3255.3
## - alcohol               1   792.02 3430.4

#model_stepwise

Explore anything highly correlated:

correlation_df <- cor(df)

#correlation_df

# fixed acidity to ph and sulphates

correlation_df_melt <- melt(correlation_df)

gz <- ggplot(correlation_df_melt, mapping = aes(x = Var1, y = Var2, fill = value)) + 
  geom_tile() + 
  theme(axis.text.x = element_text(angle = 90, hjust = 1))+
  theme(text = element_text(size = 8)) + 
  ggtitle("Heat map for correlation") + 
  ylab("")+
  xlab("")+
  scale_fill_distiller(palette = "RdPu")

ggplotly(gz, tooltip = "text")

minimized_lm_df <- df %>% 
  select(-free.sulfur.dioxide, -citric.acid, -density, -volatile.acidity, -pH)

minimized_lm <- lm(data = minimized_lm_df, quality ~ .)

summary(minimized_lm)

## 
## Call:
## lm(formula = quality ~ ., data = minimized_lm_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.71086 -0.37485 -0.06023  0.48038  2.08242 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           1.5204529  0.2113680   7.193 9.69e-13 ***
## fixed.acidity         0.0485196  0.0101132   4.798 1.76e-06 ***
## residual.sugar        0.0057865  0.0124562   0.465    0.642    
## chlorides            -2.3443435  0.4030377  -5.817 7.24e-09 ***
## total.sulfur.dioxide -0.0023418  0.0005467  -4.283 1.95e-05 ***
## sulphates             1.1850119  0.1116567  10.613  < 2e-16 ***
## alcohol               0.3100041  0.0171438  18.083  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6742 on 1592 degrees of freedom
## Multiple R-squared:  0.3057, Adjusted R-squared:  0.3031 
## F-statistic: 116.8 on 6 and 1592 DF,  p-value: < 2.2e-16

correlations

findCorrelation(
  cor(df),
  cutoff = 0.5,
  verbose = TRUE,
  names = TRUE
)

## Compare row 3  and column  1 with corr  0.672 
##   Means:  0.293 vs 0.19 so flagging column 3 
## Compare row 1  and column  8 with corr  0.668 
##   Means:  0.245 vs 0.172 so flagging column 1 
## Compare row 7  and column  6 with corr  0.668 
##   Means:  0.174 vs 0.169 so flagging column 7 
## All correlations <= 0.5

## [1] "citric.acid"          "fixed.acidity"        "total.sulfur.dioxide"

ggcorrplot(correlation_df, hc.order = TRUE, type = "lower")

df_less_acid <- df %>% 
  select(-citric.acid, -fixed.acidity)

lm_less_acid <- lm(data = df_less_acid, quality ~ .)
summary(lm_less_acid)

## 
## Call:
## lm(formula = quality ~ ., data = df_less_acid)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.66740 -0.37223 -0.04556  0.46264  2.03092 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          12.0575013 12.0089257   1.004   0.3155    
## volatile.acidity     -1.0127925  0.1009279 -10.035  < 2e-16 ***
## residual.sugar        0.0115046  0.0134606   0.855   0.3929    
## chlorides            -2.0490938  0.3992190  -5.133 3.21e-07 ***
## free.sulfur.dioxide   0.0048741  0.0021385   2.279   0.0228 *  
## total.sulfur.dioxide -0.0035687  0.0006939  -5.143 3.03e-07 ***
## density              -7.5668358 11.8648390  -0.638   0.5237    
## pH                   -0.4919908  0.1210931  -4.063 5.08e-05 ***
## sulphates             0.9024434  0.1129657   7.989 2.60e-15 ***
## alcohol               0.2810022  0.0201992  13.912  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.648 on 1589 degrees of freedom
## Multiple R-squared:  0.3598, Adjusted R-squared:  0.3562 
## F-statistic: 99.22 on 9 and 1589 DF,  p-value: < 2.2e-16

Lasso

plot(df)

Full scaled analysis

colnames(df)

##  [1] "fixed.acidity"        "volatile.acidity"     "citric.acid"         
##  [4] "residual.sugar"       "chlorides"            "free.sulfur.dioxide" 
##  [7] "total.sulfur.dioxide" "density"              "pH"                  
## [10] "sulphates"            "alcohol"              "quality"

cols = c('fixed.acidity', 'volatile.acidity', 'citric.acid', 'residual.sugar', 'chlorides', 'free.sulfur.dioxide', 'total.sulfur.dioxide', 'density', 'pH', 'sulphates', 'alcohol')

pre_proc_val <- preProcess(train[,cols], method = c("center", "scale"))

train[,cols] = predict(pre_proc_val, train[,cols])
test[,cols] = predict(pre_proc_val, test[,cols])

summary(train)

##  fixed.acidity     volatile.acidity     citric.acid       residual.sugar    
##  Min.   :-2.0849   Min.   :-2.329919   Min.   :-1.36303   Min.   :-1.16750  
##  1st Qu.:-0.6950   1st Qu.:-0.744594   1st Qu.:-0.90785   1st Qu.:-0.46897  
##  Median :-0.2317   Median :-0.008551   Median :-0.09862   Median :-0.25941  
##  Mean   : 0.0000   Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.00000  
##  3rd Qu.: 0.5212   3rd Qu.: 0.614255   3rd Qu.: 0.81176   3rd Qu.: 0.02001  
##  Max.   : 4.2278   Max.   : 5.936415   Max.   : 3.69462   Max.   : 9.03108  
##    chlorides        free.sulfur.dioxide total.sulfur.dioxide
##  Min.   :-1.53614   Min.   :-1.4285     Min.   :-1.2448     
##  1st Qu.:-0.36667   1st Qu.:-0.7553     1st Qu.:-0.7563     
##  Median :-0.16504   Median :-0.2744     Median :-0.2678     
##  Mean   : 0.00000   Mean   : 0.0000     Mean   : 0.0000     
##  3rd Qu.: 0.05676   3rd Qu.: 0.4950     3rd Qu.: 0.4955     
##  Max.   :10.54169   Max.   : 5.0152     Max.   : 7.3955     
##     density                pH             sulphates          alcohol       
##  Min.   :-3.517805   Min.   :-3.66910   Min.   :-1.9481   Min.   :-1.9002  
##  1st Qu.:-0.608370   1st Qu.:-0.66021   1st Qu.:-0.6163   1st Qu.:-0.8625  
##  Median :-0.003333   Median :-0.02002   Median :-0.2316   Median :-0.2965  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.588550   3rd Qu.: 0.55615   3rd Qu.: 0.3603   3rd Qu.: 0.5840  
##  Max.   : 3.647929   Max.   : 4.46130   Max.   : 7.9365   Max.   : 3.3827  
##     quality     
##  Min.   :3.000  
##  1st Qu.:5.000  
##  Median :6.000  
##  Mean   :5.624  
##  3rd Qu.:6.000  
##  Max.   :8.000

#generic LM

lm_2 <- lm(quality ~ ., data = train)
summary(lm_2)

## 
## Call:
## lm(formula = quality ~ ., data = train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.2357 -0.3565 -0.0559  0.4114  2.0607 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           5.62377    0.01904 295.309  < 2e-16 ***
## fixed.acidity         0.08870    0.05479   1.619 0.105768    
## volatile.acidity     -0.19879    0.02547  -7.805 1.37e-14 ***
## citric.acid          -0.03644    0.03361  -1.084 0.278593    
## residual.sugar        0.03404    0.02532   1.344 0.179089    
## chlorides            -0.08550    0.02384  -3.587 0.000349 ***
## free.sulfur.dioxide   0.02978    0.02640   1.128 0.259610    
## total.sulfur.dioxide -0.10307    0.02777  -3.711 0.000217 ***
## density              -0.04691    0.04896  -0.958 0.338150    
## pH                   -0.02911    0.03574  -0.815 0.415470    
## sulphates             0.14075    0.02292   6.141 1.14e-09 ***
## alcohol               0.28610    0.03272   8.744  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.637 on 1107 degrees of freedom
## Multiple R-squared:  0.3642, Adjusted R-squared:  0.3579 
## F-statistic: 57.65 on 11 and 1107 DF,  p-value: < 2.2e-16

Regularization

cols_reg = c('fixed.acidity', 'volatile.acidity', 'citric.acid', 'residual.sugar', 'chlorides', 'free.sulfur.dioxide', 'total.sulfur.dioxide', 'density', 'pH', 'sulphates', 'alcohol', 'quality')

dummies <- dummyVars(quality ~ ., data = df[,cols_reg])

train_dummies = predict(dummies, newdata = train[,cols_reg])

test_dummies = predict(dummies, newdata = test[,cols_reg])

print(dim(train_dummies)); print(dim(test_dummies))

## [1] 1119   11

## [1] 480  11

Ridge Regression

x = as.matrix(train_dummies)
y_train = train$quality

x_test = as.matrix(test_dummies)
y_test = test$quality

lambdas <- 10^seq(2, -3, by = -.1)
ridge_reg = glmnet(x, y_train, nlambda = 25, alpha = 0, family = 'gaussian', lambda = lambdas)

summary(ridge_reg)

##           Length Class     Mode   
## a0         51    -none-    numeric
## beta      561    dgCMatrix S4     
## df         51    -none-    numeric
## dim         2    -none-    numeric
## lambda     51    -none-    numeric
## dev.ratio  51    -none-    numeric
## nulldev     1    -none-    numeric
## npasses     1    -none-    numeric
## jerr        1    -none-    numeric
## offset      1    -none-    logical
## call        7    -none-    call   
## nobs        1    -none-    numeric

Optimal Lambda

cv_ridge <- cv.glmnet(x, y_train, alpha = 0, lambda = lambdas)
optimal_lambda <- cv_ridge$lambda.min
optimal_lambda

## [1] 0.03162278

Formal metrics for Ridge

# Compute R^2 from true and predicted values
eval_results <- function(true, predicted, df) {
  SSE <- sum((predicted - true)^2)
  SST <- sum((true - mean(true))^2)
  R_square <- 1 - SSE / SST
  RMSE = sqrt(SSE/nrow(df))

  
  # Model performance metrics
data.frame(
  RMSE = RMSE,
  Rsquare = R_square
)
  
}

# Prediction and evaluation on train data
predictions_train <- predict(ridge_reg, s = optimal_lambda, newx = x)
eval_results(y_train, predictions_train, train)

##        RMSE   Rsquare
## 1 0.6338504 0.3637537

# Prediction and evaluation on test data
predictions_test <- predict(ridge_reg, s = optimal_lambda, newx = x_test)
eval_results(y_test, predictions_test, test)

##        RMSE   Rsquare
## 1 0.6757041 0.3457975

Lasso

lambdas <- 10^seq(2, -3, by = -.1)

# Setting alpha = 1 implements lasso regression
lasso_reg <- cv.glmnet(x, y_train, alpha = 1, lambda = lambdas, standardize = TRUE, nfolds = 5)

# Best 
lambda_best <- lasso_reg$lambda.min 
lambda_best

## [1] 0.01

lasso_model <- glmnet(x, y_train, alpha = 1, lambda = lambda_best, standardize = TRUE)

predictions_train <- predict(lasso_model, s = lambda_best, newx = x)
eval_results(y_train, predictions_train, train)

##        RMSE   Rsquare
## 1 0.6351857 0.3610703

predictions_test <- predict(lasso_model, s = lambda_best, newx = x_test)
eval_results(y_test, predictions_test, test)

##        RMSE   Rsquare
## 1 0.6775253 0.3422662

Mix of ridge and lasso (elasticnet)

# Set training control
train_cont <- trainControl(method = "repeatedcv",
                              number = 10,
                              repeats = 5,
                              search = "random",
                              verboseIter = TRUE)

# Train the model
elastic_reg <- train(quality ~ .,
                           data = train,
                           method = "glmnet",
                           preProcess = c("center", "scale"),
                           tuneLength = 10,
                           trControl = train_cont)

## + Fold01.Rep1: alpha=0.62989, lambda=0.003636 
## - Fold01.Rep1: alpha=0.62989, lambda=0.003636 
## + Fold01.Rep1: alpha=0.29776, lambda=2.394057 
## - Fold01.Rep1: alpha=0.29776, lambda=2.394057 
## + Fold01.Rep1: alpha=0.28799, lambda=0.824474 
## - Fold01.Rep1: alpha=0.28799, lambda=0.824474 
## + Fold01.Rep1: alpha=0.94218, lambda=2.531489 
## - Fold01.Rep1: alpha=0.94218, lambda=2.531489 
## + Fold01.Rep1: alpha=0.35679, lambda=1.223740 
## - Fold01.Rep1: alpha=0.35679, lambda=1.223740 
## + Fold01.Rep1: alpha=0.63956, lambda=3.222801 
## - Fold01.Rep1: alpha=0.63956, lambda=3.222801 
## + Fold01.Rep1: alpha=0.65727, lambda=0.001635 
## - Fold01.Rep1: alpha=0.65727, lambda=0.001635 
## + Fold01.Rep1: alpha=0.23136, lambda=0.011723 
## - Fold01.Rep1: alpha=0.23136, lambda=0.011723 
## + Fold01.Rep1: alpha=0.08606, lambda=0.018838 
## - Fold01.Rep1: alpha=0.08606, lambda=0.018838 
## + Fold01.Rep1: alpha=0.27970, lambda=0.518103 
## - Fold01.Rep1: alpha=0.27970, lambda=0.518103 
## + Fold02.Rep1: alpha=0.62989, lambda=0.003636 
## - Fold02.Rep1: alpha=0.62989, lambda=0.003636 
## + Fold02.Rep1: alpha=0.29776, lambda=2.394057 
## - Fold02.Rep1: alpha=0.29776, lambda=2.394057 
## + Fold02.Rep1: alpha=0.28799, lambda=0.824474 
## - Fold02.Rep1: alpha=0.28799, lambda=0.824474 
## + Fold02.Rep1: alpha=0.94218, lambda=2.531489 
## - Fold02.Rep1: alpha=0.94218, lambda=2.531489 
## + Fold02.Rep1: alpha=0.35679, lambda=1.223740 
## - Fold02.Rep1: alpha=0.35679, lambda=1.223740 
## + Fold02.Rep1: alpha=0.63956, lambda=3.222801 
## - Fold02.Rep1: alpha=0.63956, lambda=3.222801 
## + Fold02.Rep1: alpha=0.65727, lambda=0.001635 
## - Fold02.Rep1: alpha=0.65727, lambda=0.001635 
## + Fold02.Rep1: alpha=0.23136, lambda=0.011723 
## - Fold02.Rep1: alpha=0.23136, lambda=0.011723 
## + Fold02.Rep1: alpha=0.08606, lambda=0.018838 
## - Fold02.Rep1: alpha=0.08606, lambda=0.018838 
## + Fold02.Rep1: alpha=0.27970, lambda=0.518103 
## - Fold02.Rep1: alpha=0.27970, lambda=0.518103 
## + Fold03.Rep1: alpha=0.62989, lambda=0.003636 
## - Fold03.Rep1: alpha=0.62989, lambda=0.003636 
## + Fold03.Rep1: alpha=0.29776, lambda=2.394057 
## - Fold03.Rep1: alpha=0.29776, lambda=2.394057 
## + Fold03.Rep1: alpha=0.28799, lambda=0.824474 
## - Fold03.Rep1: alpha=0.28799, lambda=0.824474 
## + Fold03.Rep1: alpha=0.94218, lambda=2.531489 
## - Fold03.Rep1: alpha=0.94218, lambda=2.531489 
## + Fold03.Rep1: alpha=0.35679, lambda=1.223740 
## - Fold03.Rep1: alpha=0.35679, lambda=1.223740 
## + Fold03.Rep1: alpha=0.63956, lambda=3.222801 
## - Fold03.Rep1: alpha=0.63956, lambda=3.222801 
## + Fold03.Rep1: alpha=0.65727, lambda=0.001635 
## - Fold03.Rep1: alpha=0.65727, lambda=0.001635 
## + Fold03.Rep1: alpha=0.23136, lambda=0.011723 
## - Fold03.Rep1: alpha=0.23136, lambda=0.011723 
## + Fold03.Rep1: alpha=0.08606, lambda=0.018838 
## - Fold03.Rep1: alpha=0.08606, lambda=0.018838 
## + Fold03.Rep1: alpha=0.27970, lambda=0.518103 
## - Fold03.Rep1: alpha=0.27970, lambda=0.518103 
## + Fold04.Rep1: alpha=0.62989, lambda=0.003636 
## - Fold04.Rep1: alpha=0.62989, lambda=0.003636 
## + Fold04.Rep1: alpha=0.29776, lambda=2.394057 
## - Fold04.Rep1: alpha=0.29776, lambda=2.394057 
## + Fold04.Rep1: alpha=0.28799, lambda=0.824474 
## - Fold04.Rep1: alpha=0.28799, lambda=0.824474 
## + Fold04.Rep1: alpha=0.94218, lambda=2.531489 
## - Fold04.Rep1: alpha=0.94218, lambda=2.531489 
## + Fold04.Rep1: alpha=0.35679, lambda=1.223740 
## - Fold04.Rep1: alpha=0.35679, lambda=1.223740 
## + Fold04.Rep1: alpha=0.63956, lambda=3.222801 
## - Fold04.Rep1: alpha=0.63956, lambda=3.222801 
## + Fold04.Rep1: alpha=0.65727, lambda=0.001635 
## - Fold04.Rep1: alpha=0.65727, lambda=0.001635 
## + Fold04.Rep1: alpha=0.23136, lambda=0.011723 
## - Fold04.Rep1: alpha=0.23136, lambda=0.011723 
## + Fold04.Rep1: alpha=0.08606, lambda=0.018838 
## - Fold04.Rep1: alpha=0.08606, lambda=0.018838 
## + Fold04.Rep1: alpha=0.27970, lambda=0.518103 
## - Fold04.Rep1: alpha=0.27970, lambda=0.518103 
## + Fold05.Rep1: alpha=0.62989, lambda=0.003636 
## - Fold05.Rep1: alpha=0.62989, lambda=0.003636 
## + Fold05.Rep1: alpha=0.29776, lambda=2.394057 
## - Fold05.Rep1: alpha=0.29776, lambda=2.394057 
## + Fold05.Rep1: alpha=0.28799, lambda=0.824474 
## - Fold05.Rep1: alpha=0.28799, lambda=0.824474 
## + Fold05.Rep1: alpha=0.94218, lambda=2.531489 
## - Fold05.Rep1: alpha=0.94218, lambda=2.531489 
## + Fold05.Rep1: alpha=0.35679, lambda=1.223740 
## - Fold05.Rep1: alpha=0.35679, lambda=1.223740 
## + Fold05.Rep1: alpha=0.63956, lambda=3.222801 
## - Fold05.Rep1: alpha=0.63956, lambda=3.222801 
## + Fold05.Rep1: alpha=0.65727, lambda=0.001635 
## - Fold05.Rep1: alpha=0.65727, lambda=0.001635 
## + Fold05.Rep1: alpha=0.23136, lambda=0.011723 
## - Fold05.Rep1: alpha=0.23136, lambda=0.011723 
## + Fold05.Rep1: alpha=0.08606, lambda=0.018838 
## - Fold05.Rep1: alpha=0.08606, lambda=0.018838 
## + Fold05.Rep1: alpha=0.27970, lambda=0.518103 
## - Fold05.Rep1: alpha=0.27970, lambda=0.518103 
## + Fold06.Rep1: alpha=0.62989, lambda=0.003636 
## - Fold06.Rep1: alpha=0.62989, lambda=0.003636 
## + Fold06.Rep1: alpha=0.29776, lambda=2.394057 
## - Fold06.Rep1: alpha=0.29776, lambda=2.394057 
## + Fold06.Rep1: alpha=0.28799, lambda=0.824474 
## - Fold06.Rep1: alpha=0.28799, lambda=0.824474 
## + Fold06.Rep1: alpha=0.94218, lambda=2.531489 
## - Fold06.Rep1: alpha=0.94218, lambda=2.531489 
## + Fold06.Rep1: alpha=0.35679, lambda=1.223740 
## - Fold06.Rep1: alpha=0.35679, lambda=1.223740 
## + Fold06.Rep1: alpha=0.63956, lambda=3.222801 
## - Fold06.Rep1: alpha=0.63956, lambda=3.222801 
## + Fold06.Rep1: alpha=0.65727, lambda=0.001635 
## - Fold06.Rep1: alpha=0.65727, lambda=0.001635 
## + Fold06.Rep1: alpha=0.23136, lambda=0.011723 
## - Fold06.Rep1: alpha=0.23136, lambda=0.011723 
## + Fold06.Rep1: alpha=0.08606, lambda=0.018838 
## - Fold06.Rep1: alpha=0.08606, lambda=0.018838 
## + Fold06.Rep1: alpha=0.27970, lambda=0.518103 
## - Fold06.Rep1: alpha=0.27970, lambda=0.518103 
## + Fold07.Rep1: alpha=0.62989, lambda=0.003636 
## - Fold07.Rep1: alpha=0.62989, lambda=0.003636 
## + Fold07.Rep1: alpha=0.29776, lambda=2.394057 
## - Fold07.Rep1: alpha=0.29776, lambda=2.394057 
## + Fold07.Rep1: alpha=0.28799, lambda=0.824474 
## - Fold07.Rep1: alpha=0.28799, lambda=0.824474 
## + Fold07.Rep1: alpha=0.94218, lambda=2.531489 
## - Fold07.Rep1: alpha=0.94218, lambda=2.531489 
## + Fold07.Rep1: alpha=0.35679, lambda=1.223740 
## - Fold07.Rep1: alpha=0.35679, lambda=1.223740 
## + Fold07.Rep1: alpha=0.63956, lambda=3.222801 
## - Fold07.Rep1: alpha=0.63956, lambda=3.222801 
## + Fold07.Rep1: alpha=0.65727, lambda=0.001635 
## - Fold07.Rep1: alpha=0.65727, lambda=0.001635 
## + Fold07.Rep1: alpha=0.23136, lambda=0.011723 
## - Fold07.Rep1: alpha=0.23136, lambda=0.011723 
## + Fold07.Rep1: alpha=0.08606, lambda=0.018838 
## - Fold07.Rep1: alpha=0.08606, lambda=0.018838 
## + Fold07.Rep1: alpha=0.27970, lambda=0.518103 
## - Fold07.Rep1: alpha=0.27970, lambda=0.518103 
## + Fold08.Rep1: alpha=0.62989, lambda=0.003636 
## - Fold08.Rep1: alpha=0.62989, lambda=0.003636 
## + Fold08.Rep1: alpha=0.29776, lambda=2.394057 
## - Fold08.Rep1: alpha=0.29776, lambda=2.394057 
## + Fold08.Rep1: alpha=0.28799, lambda=0.824474 
## - Fold08.Rep1: alpha=0.28799, lambda=0.824474 
## + Fold08.Rep1: alpha=0.94218, lambda=2.531489 
## - Fold08.Rep1: alpha=0.94218, lambda=2.531489 
## + Fold08.Rep1: alpha=0.35679, lambda=1.223740 
## - Fold08.Rep1: alpha=0.35679, lambda=1.223740 
## + Fold08.Rep1: alpha=0.63956, lambda=3.222801 
## - Fold08.Rep1: alpha=0.63956, lambda=3.222801 
## + Fold08.Rep1: alpha=0.65727, lambda=0.001635 
## - Fold08.Rep1: alpha=0.65727, lambda=0.001635 
## + Fold08.Rep1: alpha=0.23136, lambda=0.011723 
## - Fold08.Rep1: alpha=0.23136, lambda=0.011723 
## + Fold08.Rep1: alpha=0.08606, lambda=0.018838 
## - Fold08.Rep1: alpha=0.08606, lambda=0.018838 
## + Fold08.Rep1: alpha=0.27970, lambda=0.518103 
## - Fold08.Rep1: alpha=0.27970, lambda=0.518103 
## + Fold09.Rep1: alpha=0.62989, lambda=0.003636 
## - Fold09.Rep1: alpha=0.62989, lambda=0.003636 
## + Fold09.Rep1: alpha=0.29776, lambda=2.394057 
## - Fold09.Rep1: alpha=0.29776, lambda=2.394057 
## + Fold09.Rep1: alpha=0.28799, lambda=0.824474 
## - Fold09.Rep1: alpha=0.28799, lambda=0.824474 
## + Fold09.Rep1: alpha=0.94218, lambda=2.531489 
## - Fold09.Rep1: alpha=0.94218, lambda=2.531489 
## + Fold09.Rep1: alpha=0.35679, lambda=1.223740 
## - Fold09.Rep1: alpha=0.35679, lambda=1.223740 
## + Fold09.Rep1: alpha=0.63956, lambda=3.222801 
## - Fold09.Rep1: alpha=0.63956, lambda=3.222801 
## + Fold09.Rep1: alpha=0.65727, lambda=0.001635 
## - Fold09.Rep1: alpha=0.65727, lambda=0.001635 
## + Fold09.Rep1: alpha=0.23136, lambda=0.011723 
## - Fold09.Rep1: alpha=0.23136, lambda=0.011723 
## + Fold09.Rep1: alpha=0.08606, lambda=0.018838 
## - Fold09.Rep1: alpha=0.08606, lambda=0.018838 
## + Fold09.Rep1: alpha=0.27970, lambda=0.518103 
## - Fold09.Rep1: alpha=0.27970, lambda=0.518103 
## + Fold10.Rep1: alpha=0.62989, lambda=0.003636 
## - Fold10.Rep1: alpha=0.62989, lambda=0.003636 
## + Fold10.Rep1: alpha=0.29776, lambda=2.394057 
## - Fold10.Rep1: alpha=0.29776, lambda=2.394057 
## + Fold10.Rep1: alpha=0.28799, lambda=0.824474 
## - Fold10.Rep1: alpha=0.28799, lambda=0.824474 
## + Fold10.Rep1: alpha=0.94218, lambda=2.531489 
## - Fold10.Rep1: alpha=0.94218, lambda=2.531489 
## + Fold10.Rep1: alpha=0.35679, lambda=1.223740 
## - Fold10.Rep1: alpha=0.35679, lambda=1.223740 
## + Fold10.Rep1: alpha=0.63956, lambda=3.222801 
## - Fold10.Rep1: alpha=0.63956, lambda=3.222801 
## + Fold10.Rep1: alpha=0.65727, lambda=0.001635 
## - Fold10.Rep1: alpha=0.65727, lambda=0.001635 
## + Fold10.Rep1: alpha=0.23136, lambda=0.011723 
## - Fold10.Rep1: alpha=0.23136, lambda=0.011723 
## + Fold10.Rep1: alpha=0.08606, lambda=0.018838 
## - Fold10.Rep1: alpha=0.08606, lambda=0.018838 
## + Fold10.Rep1: alpha=0.27970, lambda=0.518103 
## - Fold10.Rep1: alpha=0.27970, lambda=0.518103 
## + Fold01.Rep2: alpha=0.62989, lambda=0.003636 
## - Fold01.Rep2: alpha=0.62989, lambda=0.003636 
## + Fold01.Rep2: alpha=0.29776, lambda=2.394057 
## - Fold01.Rep2: alpha=0.29776, lambda=2.394057 
## + Fold01.Rep2: alpha=0.28799, lambda=0.824474 
## - Fold01.Rep2: alpha=0.28799, lambda=0.824474 
## + Fold01.Rep2: alpha=0.94218, lambda=2.531489 
## - Fold01.Rep2: alpha=0.94218, lambda=2.531489 
## + Fold01.Rep2: alpha=0.35679, lambda=1.223740 
## - Fold01.Rep2: alpha=0.35679, lambda=1.223740 
## + Fold01.Rep2: alpha=0.63956, lambda=3.222801 
## - Fold01.Rep2: alpha=0.63956, lambda=3.222801 
## + Fold01.Rep2: alpha=0.65727, lambda=0.001635 
## - Fold01.Rep2: alpha=0.65727, lambda=0.001635 
## + Fold01.Rep2: alpha=0.23136, lambda=0.011723 
## - Fold01.Rep2: alpha=0.23136, lambda=0.011723 
## + Fold01.Rep2: alpha=0.08606, lambda=0.018838 
## - Fold01.Rep2: alpha=0.08606, lambda=0.018838 
## + Fold01.Rep2: alpha=0.27970, lambda=0.518103 
## - Fold01.Rep2: alpha=0.27970, lambda=0.518103 
## + Fold02.Rep2: alpha=0.62989, lambda=0.003636 
## - Fold02.Rep2: alpha=0.62989, lambda=0.003636 
## + Fold02.Rep2: alpha=0.29776, lambda=2.394057 
## - Fold02.Rep2: alpha=0.29776, lambda=2.394057 
## + Fold02.Rep2: alpha=0.28799, lambda=0.824474 
## - Fold02.Rep2: alpha=0.28799, lambda=0.824474 
## + Fold02.Rep2: alpha=0.94218, lambda=2.531489 
## - Fold02.Rep2: alpha=0.94218, lambda=2.531489 
## + Fold02.Rep2: alpha=0.35679, lambda=1.223740 
## - Fold02.Rep2: alpha=0.35679, lambda=1.223740 
## + Fold02.Rep2: alpha=0.63956, lambda=3.222801 
## - Fold02.Rep2: alpha=0.63956, lambda=3.222801 
## + Fold02.Rep2: alpha=0.65727, lambda=0.001635 
## - Fold02.Rep2: alpha=0.65727, lambda=0.001635 
## + Fold02.Rep2: alpha=0.23136, lambda=0.011723 
## - Fold02.Rep2: alpha=0.23136, lambda=0.011723 
## + Fold02.Rep2: alpha=0.08606, lambda=0.018838 
## - Fold02.Rep2: alpha=0.08606, lambda=0.018838 
## + Fold02.Rep2: alpha=0.27970, lambda=0.518103 
## - Fold02.Rep2: alpha=0.27970, lambda=0.518103 
## + Fold03.Rep2: alpha=0.62989, lambda=0.003636 
## - Fold03.Rep2: alpha=0.62989, lambda=0.003636 
## + Fold03.Rep2: alpha=0.29776, lambda=2.394057 
## - Fold03.Rep2: alpha=0.29776, lambda=2.394057 
## + Fold03.Rep2: alpha=0.28799, lambda=0.824474 
## - Fold03.Rep2: alpha=0.28799, lambda=0.824474 
## + Fold03.Rep2: alpha=0.94218, lambda=2.531489 
## - Fold03.Rep2: alpha=0.94218, lambda=2.531489 
## + Fold03.Rep2: alpha=0.35679, lambda=1.223740 
## - Fold03.Rep2: alpha=0.35679, lambda=1.223740 
## + Fold03.Rep2: alpha=0.63956, lambda=3.222801 
## - Fold03.Rep2: alpha=0.63956, lambda=3.222801 
## + Fold03.Rep2: alpha=0.65727, lambda=0.001635 
## - Fold03.Rep2: alpha=0.65727, lambda=0.001635 
## + Fold03.Rep2: alpha=0.23136, lambda=0.011723 
## - Fold03.Rep2: alpha=0.23136, lambda=0.011723 
## + Fold03.Rep2: alpha=0.08606, lambda=0.018838 
## - Fold03.Rep2: alpha=0.08606, lambda=0.018838 
## + Fold03.Rep2: alpha=0.27970, lambda=0.518103 
## - Fold03.Rep2: alpha=0.27970, lambda=0.518103 
## + Fold04.Rep2: alpha=0.62989, lambda=0.003636 
## - Fold04.Rep2: alpha=0.62989, lambda=0.003636 
## + Fold04.Rep2: alpha=0.29776, lambda=2.394057 
## - Fold04.Rep2: alpha=0.29776, lambda=2.394057 
## + Fold04.Rep2: alpha=0.28799, lambda=0.824474 
## - Fold04.Rep2: alpha=0.28799, lambda=0.824474 
## + Fold04.Rep2: alpha=0.94218, lambda=2.531489 
## - Fold04.Rep2: alpha=0.94218, lambda=2.531489 
## + Fold04.Rep2: alpha=0.35679, lambda=1.223740 
## - Fold04.Rep2: alpha=0.35679, lambda=1.223740 
## + Fold04.Rep2: alpha=0.63956, lambda=3.222801 
## - Fold04.Rep2: alpha=0.63956, lambda=3.222801 
## + Fold04.Rep2: alpha=0.65727, lambda=0.001635 
## - Fold04.Rep2: alpha=0.65727, lambda=0.001635 
## + Fold04.Rep2: alpha=0.23136, lambda=0.011723 
## - Fold04.Rep2: alpha=0.23136, lambda=0.011723 
## + Fold04.Rep2: alpha=0.08606, lambda=0.018838 
## - Fold04.Rep2: alpha=0.08606, lambda=0.018838 
## + Fold04.Rep2: alpha=0.27970, lambda=0.518103 
## - Fold04.Rep2: alpha=0.27970, lambda=0.518103 
## + Fold05.Rep2: alpha=0.62989, lambda=0.003636 
## - Fold05.Rep2: alpha=0.62989, lambda=0.003636 
## + Fold05.Rep2: alpha=0.29776, lambda=2.394057 
## - Fold05.Rep2: alpha=0.29776, lambda=2.394057 
## + Fold05.Rep2: alpha=0.28799, lambda=0.824474 
## - Fold05.Rep2: alpha=0.28799, lambda=0.824474 
## + Fold05.Rep2: alpha=0.94218, lambda=2.531489 
## - Fold05.Rep2: alpha=0.94218, lambda=2.531489 
## + Fold05.Rep2: alpha=0.35679, lambda=1.223740 
## - Fold05.Rep2: alpha=0.35679, lambda=1.223740 
## + Fold05.Rep2: alpha=0.63956, lambda=3.222801 
## - Fold05.Rep2: alpha=0.63956, lambda=3.222801 
## + Fold05.Rep2: alpha=0.65727, lambda=0.001635 
## - Fold05.Rep2: alpha=0.65727, lambda=0.001635 
## + Fold05.Rep2: alpha=0.23136, lambda=0.011723 
## - Fold05.Rep2: alpha=0.23136, lambda=0.011723 
## + Fold05.Rep2: alpha=0.08606, lambda=0.018838 
## - Fold05.Rep2: alpha=0.08606, lambda=0.018838 
## + Fold05.Rep2: alpha=0.27970, lambda=0.518103 
## - Fold05.Rep2: alpha=0.27970, lambda=0.518103 
## + Fold06.Rep2: alpha=0.62989, lambda=0.003636 
## - Fold06.Rep2: alpha=0.62989, lambda=0.003636 
## + Fold06.Rep2: alpha=0.29776, lambda=2.394057 
## - Fold06.Rep2: alpha=0.29776, lambda=2.394057 
## + Fold06.Rep2: alpha=0.28799, lambda=0.824474 
## - Fold06.Rep2: alpha=0.28799, lambda=0.824474 
## + Fold06.Rep2: alpha=0.94218, lambda=2.531489 
## - Fold06.Rep2: alpha=0.94218, lambda=2.531489 
## + Fold06.Rep2: alpha=0.35679, lambda=1.223740 
## - Fold06.Rep2: alpha=0.35679, lambda=1.223740 
## + Fold06.Rep2: alpha=0.63956, lambda=3.222801 
## - Fold06.Rep2: alpha=0.63956, lambda=3.222801 
## + Fold06.Rep2: alpha=0.65727, lambda=0.001635 
## - Fold06.Rep2: alpha=0.65727, lambda=0.001635 
## + Fold06.Rep2: alpha=0.23136, lambda=0.011723 
## - Fold06.Rep2: alpha=0.23136, lambda=0.011723 
## + Fold06.Rep2: alpha=0.08606, lambda=0.018838 
## - Fold06.Rep2: alpha=0.08606, lambda=0.018838 
## + Fold06.Rep2: alpha=0.27970, lambda=0.518103 
## - Fold06.Rep2: alpha=0.27970, lambda=0.518103 
## + Fold07.Rep2: alpha=0.62989, lambda=0.003636 
## - Fold07.Rep2: alpha=0.62989, lambda=0.003636 
## + Fold07.Rep2: alpha=0.29776, lambda=2.394057 
## - Fold07.Rep2: alpha=0.29776, lambda=2.394057 
## + Fold07.Rep2: alpha=0.28799, lambda=0.824474 
## - Fold07.Rep2: alpha=0.28799, lambda=0.824474 
## + Fold07.Rep2: alpha=0.94218, lambda=2.531489 
## - Fold07.Rep2: alpha=0.94218, lambda=2.531489 
## + Fold07.Rep2: alpha=0.35679, lambda=1.223740 
## - Fold07.Rep2: alpha=0.35679, lambda=1.223740 
## + Fold07.Rep2: alpha=0.63956, lambda=3.222801 
## - Fold07.Rep2: alpha=0.63956, lambda=3.222801 
## + Fold07.Rep2: alpha=0.65727, lambda=0.001635 
## - Fold07.Rep2: alpha=0.65727, lambda=0.001635 
## + Fold07.Rep2: alpha=0.23136, lambda=0.011723 
## - Fold07.Rep2: alpha=0.23136, lambda=0.011723 
## + Fold07.Rep2: alpha=0.08606, lambda=0.018838 
## - Fold07.Rep2: alpha=0.08606, lambda=0.018838 
## + Fold07.Rep2: alpha=0.27970, lambda=0.518103 
## - Fold07.Rep2: alpha=0.27970, lambda=0.518103 
## + Fold08.Rep2: alpha=0.62989, lambda=0.003636 
## - Fold08.Rep2: alpha=0.62989, lambda=0.003636 
## + Fold08.Rep2: alpha=0.29776, lambda=2.394057 
## - Fold08.Rep2: alpha=0.29776, lambda=2.394057 
## + Fold08.Rep2: alpha=0.28799, lambda=0.824474 
## - Fold08.Rep2: alpha=0.28799, lambda=0.824474 
## + Fold08.Rep2: alpha=0.94218, lambda=2.531489 
## - Fold08.Rep2: alpha=0.94218, lambda=2.531489 
## + Fold08.Rep2: alpha=0.35679, lambda=1.223740 
## - Fold08.Rep2: alpha=0.35679, lambda=1.223740 
## + Fold08.Rep2: alpha=0.63956, lambda=3.222801 
## - Fold08.Rep2: alpha=0.63956, lambda=3.222801 
## + Fold08.Rep2: alpha=0.65727, lambda=0.001635 
## - Fold08.Rep2: alpha=0.65727, lambda=0.001635 
## + Fold08.Rep2: alpha=0.23136, lambda=0.011723 
## - Fold08.Rep2: alpha=0.23136, lambda=0.011723 
## + Fold08.Rep2: alpha=0.08606, lambda=0.018838 
## - Fold08.Rep2: alpha=0.08606, lambda=0.018838 
## + Fold08.Rep2: alpha=0.27970, lambda=0.518103 
## - Fold08.Rep2: alpha=0.27970, lambda=0.518103 
## + Fold09.Rep2: alpha=0.62989, lambda=0.003636 
## - Fold09.Rep2: alpha=0.62989, lambda=0.003636 
## + Fold09.Rep2: alpha=0.29776, lambda=2.394057 
## - Fold09.Rep2: alpha=0.29776, lambda=2.394057 
## + Fold09.Rep2: alpha=0.28799, lambda=0.824474 
## - Fold09.Rep2: alpha=0.28799, lambda=0.824474 
## + Fold09.Rep2: alpha=0.94218, lambda=2.531489 
## - Fold09.Rep2: alpha=0.94218, lambda=2.531489 
## + Fold09.Rep2: alpha=0.35679, lambda=1.223740 
## - Fold09.Rep2: alpha=0.35679, lambda=1.223740 
## + Fold09.Rep2: alpha=0.63956, lambda=3.222801 
## - Fold09.Rep2: alpha=0.63956, lambda=3.222801 
## + Fold09.Rep2: alpha=0.65727, lambda=0.001635 
## - Fold09.Rep2: alpha=0.65727, lambda=0.001635 
## + Fold09.Rep2: alpha=0.23136, lambda=0.011723 
## - Fold09.Rep2: alpha=0.23136, lambda=0.011723 
## + Fold09.Rep2: alpha=0.08606, lambda=0.018838 
## - Fold09.Rep2: alpha=0.08606, lambda=0.018838 
## + Fold09.Rep2: alpha=0.27970, lambda=0.518103 
## - Fold09.Rep2: alpha=0.27970, lambda=0.518103 
## + Fold10.Rep2: alpha=0.62989, lambda=0.003636 
## - Fold10.Rep2: alpha=0.62989, lambda=0.003636 
## + Fold10.Rep2: alpha=0.29776, lambda=2.394057 
## - Fold10.Rep2: alpha=0.29776, lambda=2.394057 
## + Fold10.Rep2: alpha=0.28799, lambda=0.824474 
## - Fold10.Rep2: alpha=0.28799, lambda=0.824474 
## + Fold10.Rep2: alpha=0.94218, lambda=2.531489 
## - Fold10.Rep2: alpha=0.94218, lambda=2.531489 
## + Fold10.Rep2: alpha=0.35679, lambda=1.223740 
## - Fold10.Rep2: alpha=0.35679, lambda=1.223740 
## + Fold10.Rep2: alpha=0.63956, lambda=3.222801 
## - Fold10.Rep2: alpha=0.63956, lambda=3.222801 
## + Fold10.Rep2: alpha=0.65727, lambda=0.001635 
## - Fold10.Rep2: alpha=0.65727, lambda=0.001635 
## + Fold10.Rep2: alpha=0.23136, lambda=0.011723 
## - Fold10.Rep2: alpha=0.23136, lambda=0.011723 
## + Fold10.Rep2: alpha=0.08606, lambda=0.018838 
## - Fold10.Rep2: alpha=0.08606, lambda=0.018838 
## + Fold10.Rep2: alpha=0.27970, lambda=0.518103 
## - Fold10.Rep2: alpha=0.27970, lambda=0.518103 
## + Fold01.Rep3: alpha=0.62989, lambda=0.003636 
## - Fold01.Rep3: alpha=0.62989, lambda=0.003636 
## + Fold01.Rep3: alpha=0.29776, lambda=2.394057 
## - Fold01.Rep3: alpha=0.29776, lambda=2.394057 
## + Fold01.Rep3: alpha=0.28799, lambda=0.824474 
## - Fold01.Rep3: alpha=0.28799, lambda=0.824474 
## + Fold01.Rep3: alpha=0.94218, lambda=2.531489 
## - Fold01.Rep3: alpha=0.94218, lambda=2.531489 
## + Fold01.Rep3: alpha=0.35679, lambda=1.223740 
## - Fold01.Rep3: alpha=0.35679, lambda=1.223740 
## + Fold01.Rep3: alpha=0.63956, lambda=3.222801 
## - Fold01.Rep3: alpha=0.63956, lambda=3.222801 
## + Fold01.Rep3: alpha=0.65727, lambda=0.001635 
## - Fold01.Rep3: alpha=0.65727, lambda=0.001635 
## + Fold01.Rep3: alpha=0.23136, lambda=0.011723 
## - Fold01.Rep3: alpha=0.23136, lambda=0.011723 
## + Fold01.Rep3: alpha=0.08606, lambda=0.018838 
## - Fold01.Rep3: alpha=0.08606, lambda=0.018838 
## + Fold01.Rep3: alpha=0.27970, lambda=0.518103 
## - Fold01.Rep3: alpha=0.27970, lambda=0.518103 
## + Fold02.Rep3: alpha=0.62989, lambda=0.003636 
## - Fold02.Rep3: alpha=0.62989, lambda=0.003636 
## + Fold02.Rep3: alpha=0.29776, lambda=2.394057 
## - Fold02.Rep3: alpha=0.29776, lambda=2.394057 
## + Fold02.Rep3: alpha=0.28799, lambda=0.824474 
## - Fold02.Rep3: alpha=0.28799, lambda=0.824474 
## + Fold02.Rep3: alpha=0.94218, lambda=2.531489 
## - Fold02.Rep3: alpha=0.94218, lambda=2.531489 
## + Fold02.Rep3: alpha=0.35679, lambda=1.223740 
## - Fold02.Rep3: alpha=0.35679, lambda=1.223740 
## + Fold02.Rep3: alpha=0.63956, lambda=3.222801 
## - Fold02.Rep3: alpha=0.63956, lambda=3.222801 
## + Fold02.Rep3: alpha=0.65727, lambda=0.001635 
## - Fold02.Rep3: alpha=0.65727, lambda=0.001635 
## + Fold02.Rep3: alpha=0.23136, lambda=0.011723 
## - Fold02.Rep3: alpha=0.23136, lambda=0.011723 
## + Fold02.Rep3: alpha=0.08606, lambda=0.018838 
## - Fold02.Rep3: alpha=0.08606, lambda=0.018838 
## + Fold02.Rep3: alpha=0.27970, lambda=0.518103 
## - Fold02.Rep3: alpha=0.27970, lambda=0.518103 
## + Fold03.Rep3: alpha=0.62989, lambda=0.003636 
## - Fold03.Rep3: alpha=0.62989, lambda=0.003636 
## + Fold03.Rep3: alpha=0.29776, lambda=2.394057 
## - Fold03.Rep3: alpha=0.29776, lambda=2.394057 
## + Fold03.Rep3: alpha=0.28799, lambda=0.824474 
## - Fold03.Rep3: alpha=0.28799, lambda=0.824474 
## + Fold03.Rep3: alpha=0.94218, lambda=2.531489 
## - Fold03.Rep3: alpha=0.94218, lambda=2.531489 
## + Fold03.Rep3: alpha=0.35679, lambda=1.223740 
## - Fold03.Rep3: alpha=0.35679, lambda=1.223740 
## + Fold03.Rep3: alpha=0.63956, lambda=3.222801 
## - Fold03.Rep3: alpha=0.63956, lambda=3.222801 
## + Fold03.Rep3: alpha=0.65727, lambda=0.001635 
## - Fold03.Rep3: alpha=0.65727, lambda=0.001635 
## + Fold03.Rep3: alpha=0.23136, lambda=0.011723 
## - Fold03.Rep3: alpha=0.23136, lambda=0.011723 
## + Fold03.Rep3: alpha=0.08606, lambda=0.018838 
## - Fold03.Rep3: alpha=0.08606, lambda=0.018838 
## + Fold03.Rep3: alpha=0.27970, lambda=0.518103 
## - Fold03.Rep3: alpha=0.27970, lambda=0.518103 
## + Fold04.Rep3: alpha=0.62989, lambda=0.003636 
## - Fold04.Rep3: alpha=0.62989, lambda=0.003636 
## + Fold04.Rep3: alpha=0.29776, lambda=2.394057 
## - Fold04.Rep3: alpha=0.29776, lambda=2.394057 
## + Fold04.Rep3: alpha=0.28799, lambda=0.824474 
## - Fold04.Rep3: alpha=0.28799, lambda=0.824474 
## + Fold04.Rep3: alpha=0.94218, lambda=2.531489 
## - Fold04.Rep3: alpha=0.94218, lambda=2.531489 
## + Fold04.Rep3: alpha=0.35679, lambda=1.223740 
## - Fold04.Rep3: alpha=0.35679, lambda=1.223740 
## + Fold04.Rep3: alpha=0.63956, lambda=3.222801 
## - Fold04.Rep3: alpha=0.63956, lambda=3.222801 
## + Fold04.Rep3: alpha=0.65727, lambda=0.001635 
## - Fold04.Rep3: alpha=0.65727, lambda=0.001635 
## + Fold04.Rep3: alpha=0.23136, lambda=0.011723 
## - Fold04.Rep3: alpha=0.23136, lambda=0.011723 
## + Fold04.Rep3: alpha=0.08606, lambda=0.018838 
## - Fold04.Rep3: alpha=0.08606, lambda=0.018838 
## + Fold04.Rep3: alpha=0.27970, lambda=0.518103 
## - Fold04.Rep3: alpha=0.27970, lambda=0.518103 
## + Fold05.Rep3: alpha=0.62989, lambda=0.003636 
## - Fold05.Rep3: alpha=0.62989, lambda=0.003636 
## + Fold05.Rep3: alpha=0.29776, lambda=2.394057 
## - Fold05.Rep3: alpha=0.29776, lambda=2.394057 
## + Fold05.Rep3: alpha=0.28799, lambda=0.824474 
## - Fold05.Rep3: alpha=0.28799, lambda=0.824474 
## + Fold05.Rep3: alpha=0.94218, lambda=2.531489 
## - Fold05.Rep3: alpha=0.94218, lambda=2.531489 
## + Fold05.Rep3: alpha=0.35679, lambda=1.223740 
## - Fold05.Rep3: alpha=0.35679, lambda=1.223740 
## + Fold05.Rep3: alpha=0.63956, lambda=3.222801 
## - Fold05.Rep3: alpha=0.63956, lambda=3.222801 
## + Fold05.Rep3: alpha=0.65727, lambda=0.001635 
## - Fold05.Rep3: alpha=0.65727, lambda=0.001635 
## + Fold05.Rep3: alpha=0.23136, lambda=0.011723 
## - Fold05.Rep3: alpha=0.23136, lambda=0.011723 
## + Fold05.Rep3: alpha=0.08606, lambda=0.018838 
## - Fold05.Rep3: alpha=0.08606, lambda=0.018838 
## + Fold05.Rep3: alpha=0.27970, lambda=0.518103 
## - Fold05.Rep3: alpha=0.27970, lambda=0.518103 
## + Fold06.Rep3: alpha=0.62989, lambda=0.003636 
## - Fold06.Rep3: alpha=0.62989, lambda=0.003636 
## + Fold06.Rep3: alpha=0.29776, lambda=2.394057 
## - Fold06.Rep3: alpha=0.29776, lambda=2.394057 
## + Fold06.Rep3: alpha=0.28799, lambda=0.824474 
## - Fold06.Rep3: alpha=0.28799, lambda=0.824474 
## + Fold06.Rep3: alpha=0.94218, lambda=2.531489 
## - Fold06.Rep3: alpha=0.94218, lambda=2.531489 
## + Fold06.Rep3: alpha=0.35679, lambda=1.223740 
## - Fold06.Rep3: alpha=0.35679, lambda=1.223740 
## + Fold06.Rep3: alpha=0.63956, lambda=3.222801 
## - Fold06.Rep3: alpha=0.63956, lambda=3.222801 
## + Fold06.Rep3: alpha=0.65727, lambda=0.001635 
## - Fold06.Rep3: alpha=0.65727, lambda=0.001635 
## + Fold06.Rep3: alpha=0.23136, lambda=0.011723 
## - Fold06.Rep3: alpha=0.23136, lambda=0.011723 
## + Fold06.Rep3: alpha=0.08606, lambda=0.018838 
## - Fold06.Rep3: alpha=0.08606, lambda=0.018838 
## + Fold06.Rep3: alpha=0.27970, lambda=0.518103 
## - Fold06.Rep3: alpha=0.27970, lambda=0.518103 
## + Fold07.Rep3: alpha=0.62989, lambda=0.003636 
## - Fold07.Rep3: alpha=0.62989, lambda=0.003636 
## + Fold07.Rep3: alpha=0.29776, lambda=2.394057 
## - Fold07.Rep3: alpha=0.29776, lambda=2.394057 
## + Fold07.Rep3: alpha=0.28799, lambda=0.824474 
## - Fold07.Rep3: alpha=0.28799, lambda=0.824474 
## + Fold07.Rep3: alpha=0.94218, lambda=2.531489 
## - Fold07.Rep3: alpha=0.94218, lambda=2.531489 
## + Fold07.Rep3: alpha=0.35679, lambda=1.223740 
## - Fold07.Rep3: alpha=0.35679, lambda=1.223740 
## + Fold07.Rep3: alpha=0.63956, lambda=3.222801 
## - Fold07.Rep3: alpha=0.63956, lambda=3.222801 
## + Fold07.Rep3: alpha=0.65727, lambda=0.001635 
## - Fold07.Rep3: alpha=0.65727, lambda=0.001635 
## + Fold07.Rep3: alpha=0.23136, lambda=0.011723 
## - Fold07.Rep3: alpha=0.23136, lambda=0.011723 
## + Fold07.Rep3: alpha=0.08606, lambda=0.018838 
## - Fold07.Rep3: alpha=0.08606, lambda=0.018838 
## + Fold07.Rep3: alpha=0.27970, lambda=0.518103 
## - Fold07.Rep3: alpha=0.27970, lambda=0.518103 
## + Fold08.Rep3: alpha=0.62989, lambda=0.003636 
## - Fold08.Rep3: alpha=0.62989, lambda=0.003636 
## + Fold08.Rep3: alpha=0.29776, lambda=2.394057 
## - Fold08.Rep3: alpha=0.29776, lambda=2.394057 
## + Fold08.Rep3: alpha=0.28799, lambda=0.824474 
## - Fold08.Rep3: alpha=0.28799, lambda=0.824474 
## + Fold08.Rep3: alpha=0.94218, lambda=2.531489 
## - Fold08.Rep3: alpha=0.94218, lambda=2.531489 
## + Fold08.Rep3: alpha=0.35679, lambda=1.223740 
## - Fold08.Rep3: alpha=0.35679, lambda=1.223740 
## + Fold08.Rep3: alpha=0.63956, lambda=3.222801 
## - Fold08.Rep3: alpha=0.63956, lambda=3.222801 
## + Fold08.Rep3: alpha=0.65727, lambda=0.001635 
## - Fold08.Rep3: alpha=0.65727, lambda=0.001635 
## + Fold08.Rep3: alpha=0.23136, lambda=0.011723 
## - Fold08.Rep3: alpha=0.23136, lambda=0.011723 
## + Fold08.Rep3: alpha=0.08606, lambda=0.018838 
## - Fold08.Rep3: alpha=0.08606, lambda=0.018838 
## + Fold08.Rep3: alpha=0.27970, lambda=0.518103 
## - Fold08.Rep3: alpha=0.27970, lambda=0.518103 
## + Fold09.Rep3: alpha=0.62989, lambda=0.003636 
## - Fold09.Rep3: alpha=0.62989, lambda=0.003636 
## + Fold09.Rep3: alpha=0.29776, lambda=2.394057 
## - Fold09.Rep3: alpha=0.29776, lambda=2.394057 
## + Fold09.Rep3: alpha=0.28799, lambda=0.824474 
## - Fold09.Rep3: alpha=0.28799, lambda=0.824474 
## + Fold09.Rep3: alpha=0.94218, lambda=2.531489 
## - Fold09.Rep3: alpha=0.94218, lambda=2.531489 
## + Fold09.Rep3: alpha=0.35679, lambda=1.223740 
## - Fold09.Rep3: alpha=0.35679, lambda=1.223740 
## + Fold09.Rep3: alpha=0.63956, lambda=3.222801 
## - Fold09.Rep3: alpha=0.63956, lambda=3.222801 
## + Fold09.Rep3: alpha=0.65727, lambda=0.001635 
## - Fold09.Rep3: alpha=0.65727, lambda=0.001635 
## + Fold09.Rep3: alpha=0.23136, lambda=0.011723 
## - Fold09.Rep3: alpha=0.23136, lambda=0.011723 
## + Fold09.Rep3: alpha=0.08606, lambda=0.018838 
## - Fold09.Rep3: alpha=0.08606, lambda=0.018838 
## + Fold09.Rep3: alpha=0.27970, lambda=0.518103 
## - Fold09.Rep3: alpha=0.27970, lambda=0.518103 
## + Fold10.Rep3: alpha=0.62989, lambda=0.003636 
## - Fold10.Rep3: alpha=0.62989, lambda=0.003636 
## + Fold10.Rep3: alpha=0.29776, lambda=2.394057 
## - Fold10.Rep3: alpha=0.29776, lambda=2.394057 
## + Fold10.Rep3: alpha=0.28799, lambda=0.824474 
## - Fold10.Rep3: alpha=0.28799, lambda=0.824474 
## + Fold10.Rep3: alpha=0.94218, lambda=2.531489 
## - Fold10.Rep3: alpha=0.94218, lambda=2.531489 
## + Fold10.Rep3: alpha=0.35679, lambda=1.223740 
## - Fold10.Rep3: alpha=0.35679, lambda=1.223740 
## + Fold10.Rep3: alpha=0.63956, lambda=3.222801 
## - Fold10.Rep3: alpha=0.63956, lambda=3.222801 
## + Fold10.Rep3: alpha=0.65727, lambda=0.001635 
## - Fold10.Rep3: alpha=0.65727, lambda=0.001635 
## + Fold10.Rep3: alpha=0.23136, lambda=0.011723 
## - Fold10.Rep3: alpha=0.23136, lambda=0.011723 
## + Fold10.Rep3: alpha=0.08606, lambda=0.018838 
## - Fold10.Rep3: alpha=0.08606, lambda=0.018838 
## + Fold10.Rep3: alpha=0.27970, lambda=0.518103 
## - Fold10.Rep3: alpha=0.27970, lambda=0.518103 
## + Fold01.Rep4: alpha=0.62989, lambda=0.003636 
## - Fold01.Rep4: alpha=0.62989, lambda=0.003636 
## + Fold01.Rep4: alpha=0.29776, lambda=2.394057 
## - Fold01.Rep4: alpha=0.29776, lambda=2.394057 
## + Fold01.Rep4: alpha=0.28799, lambda=0.824474 
## - Fold01.Rep4: alpha=0.28799, lambda=0.824474 
## + Fold01.Rep4: alpha=0.94218, lambda=2.531489 
## - Fold01.Rep4: alpha=0.94218, lambda=2.531489 
## + Fold01.Rep4: alpha=0.35679, lambda=1.223740 
## - Fold01.Rep4: alpha=0.35679, lambda=1.223740 
## + Fold01.Rep4: alpha=0.63956, lambda=3.222801 
## - Fold01.Rep4: alpha=0.63956, lambda=3.222801 
## + Fold01.Rep4: alpha=0.65727, lambda=0.001635 
## - Fold01.Rep4: alpha=0.65727, lambda=0.001635 
## + Fold01.Rep4: alpha=0.23136, lambda=0.011723 
## - Fold01.Rep4: alpha=0.23136, lambda=0.011723 
## + Fold01.Rep4: alpha=0.08606, lambda=0.018838 
## - Fold01.Rep4: alpha=0.08606, lambda=0.018838 
## + Fold01.Rep4: alpha=0.27970, lambda=0.518103 
## - Fold01.Rep4: alpha=0.27970, lambda=0.518103 
## + Fold02.Rep4: alpha=0.62989, lambda=0.003636 
## - Fold02.Rep4: alpha=0.62989, lambda=0.003636 
## + Fold02.Rep4: alpha=0.29776, lambda=2.394057 
## - Fold02.Rep4: alpha=0.29776, lambda=2.394057 
## + Fold02.Rep4: alpha=0.28799, lambda=0.824474 
## - Fold02.Rep4: alpha=0.28799, lambda=0.824474 
## + Fold02.Rep4: alpha=0.94218, lambda=2.531489 
## - Fold02.Rep4: alpha=0.94218, lambda=2.531489 
## + Fold02.Rep4: alpha=0.35679, lambda=1.223740 
## - Fold02.Rep4: alpha=0.35679, lambda=1.223740 
## + Fold02.Rep4: alpha=0.63956, lambda=3.222801 
## - Fold02.Rep4: alpha=0.63956, lambda=3.222801 
## + Fold02.Rep4: alpha=0.65727, lambda=0.001635 
## - Fold02.Rep4: alpha=0.65727, lambda=0.001635 
## + Fold02.Rep4: alpha=0.23136, lambda=0.011723 
## - Fold02.Rep4: alpha=0.23136, lambda=0.011723 
## + Fold02.Rep4: alpha=0.08606, lambda=0.018838 
## - Fold02.Rep4: alpha=0.08606, lambda=0.018838 
## + Fold02.Rep4: alpha=0.27970, lambda=0.518103 
## - Fold02.Rep4: alpha=0.27970, lambda=0.518103 
## + Fold03.Rep4: alpha=0.62989, lambda=0.003636 
## - Fold03.Rep4: alpha=0.62989, lambda=0.003636 
## + Fold03.Rep4: alpha=0.29776, lambda=2.394057 
## - Fold03.Rep4: alpha=0.29776, lambda=2.394057 
## + Fold03.Rep4: alpha=0.28799, lambda=0.824474 
## - Fold03.Rep4: alpha=0.28799, lambda=0.824474 
## + Fold03.Rep4: alpha=0.94218, lambda=2.531489 
## - Fold03.Rep4: alpha=0.94218, lambda=2.531489 
## + Fold03.Rep4: alpha=0.35679, lambda=1.223740 
## - Fold03.Rep4: alpha=0.35679, lambda=1.223740 
## + Fold03.Rep4: alpha=0.63956, lambda=3.222801 
## - Fold03.Rep4: alpha=0.63956, lambda=3.222801 
## + Fold03.Rep4: alpha=0.65727, lambda=0.001635 
## - Fold03.Rep4: alpha=0.65727, lambda=0.001635 
## + Fold03.Rep4: alpha=0.23136, lambda=0.011723 
## - Fold03.Rep4: alpha=0.23136, lambda=0.011723 
## + Fold03.Rep4: alpha=0.08606, lambda=0.018838 
## - Fold03.Rep4: alpha=0.08606, lambda=0.018838 
## + Fold03.Rep4: alpha=0.27970, lambda=0.518103 
## - Fold03.Rep4: alpha=0.27970, lambda=0.518103 
## + Fold04.Rep4: alpha=0.62989, lambda=0.003636 
## - Fold04.Rep4: alpha=0.62989, lambda=0.003636 
## + Fold04.Rep4: alpha=0.29776, lambda=2.394057 
## - Fold04.Rep4: alpha=0.29776, lambda=2.394057 
## + Fold04.Rep4: alpha=0.28799, lambda=0.824474 
## - Fold04.Rep4: alpha=0.28799, lambda=0.824474 
## + Fold04.Rep4: alpha=0.94218, lambda=2.531489 
## - Fold04.Rep4: alpha=0.94218, lambda=2.531489 
## + Fold04.Rep4: alpha=0.35679, lambda=1.223740 
## - Fold04.Rep4: alpha=0.35679, lambda=1.223740 
## + Fold04.Rep4: alpha=0.63956, lambda=3.222801 
## - Fold04.Rep4: alpha=0.63956, lambda=3.222801 
## + Fold04.Rep4: alpha=0.65727, lambda=0.001635 
## - Fold04.Rep4: alpha=0.65727, lambda=0.001635 
## + Fold04.Rep4: alpha=0.23136, lambda=0.011723 
## - Fold04.Rep4: alpha=0.23136, lambda=0.011723 
## + Fold04.Rep4: alpha=0.08606, lambda=0.018838 
## - Fold04.Rep4: alpha=0.08606, lambda=0.018838 
## + Fold04.Rep4: alpha=0.27970, lambda=0.518103 
## - Fold04.Rep4: alpha=0.27970, lambda=0.518103 
## + Fold05.Rep4: alpha=0.62989, lambda=0.003636 
## - Fold05.Rep4: alpha=0.62989, lambda=0.003636 
## + Fold05.Rep4: alpha=0.29776, lambda=2.394057 
## - Fold05.Rep4: alpha=0.29776, lambda=2.394057 
## + Fold05.Rep4: alpha=0.28799, lambda=0.824474 
## - Fold05.Rep4: alpha=0.28799, lambda=0.824474 
## + Fold05.Rep4: alpha=0.94218, lambda=2.531489 
## - Fold05.Rep4: alpha=0.94218, lambda=2.531489 
## + Fold05.Rep4: alpha=0.35679, lambda=1.223740 
## - Fold05.Rep4: alpha=0.35679, lambda=1.223740 
## + Fold05.Rep4: alpha=0.63956, lambda=3.222801 
## - Fold05.Rep4: alpha=0.63956, lambda=3.222801 
## + Fold05.Rep4: alpha=0.65727, lambda=0.001635 
## - Fold05.Rep4: alpha=0.65727, lambda=0.001635 
## + Fold05.Rep4: alpha=0.23136, lambda=0.011723 
## - Fold05.Rep4: alpha=0.23136, lambda=0.011723 
## + Fold05.Rep4: alpha=0.08606, lambda=0.018838 
## - Fold05.Rep4: alpha=0.08606, lambda=0.018838 
## + Fold05.Rep4: alpha=0.27970, lambda=0.518103 
## - Fold05.Rep4: alpha=0.27970, lambda=0.518103 
## + Fold06.Rep4: alpha=0.62989, lambda=0.003636 
## - Fold06.Rep4: alpha=0.62989, lambda=0.003636 
## + Fold06.Rep4: alpha=0.29776, lambda=2.394057 
## - Fold06.Rep4: alpha=0.29776, lambda=2.394057 
## + Fold06.Rep4: alpha=0.28799, lambda=0.824474 
## - Fold06.Rep4: alpha=0.28799, lambda=0.824474 
## + Fold06.Rep4: alpha=0.94218, lambda=2.531489 
## - Fold06.Rep4: alpha=0.94218, lambda=2.531489 
## + Fold06.Rep4: alpha=0.35679, lambda=1.223740 
## - Fold06.Rep4: alpha=0.35679, lambda=1.223740 
## + Fold06.Rep4: alpha=0.63956, lambda=3.222801 
## - Fold06.Rep4: alpha=0.63956, lambda=3.222801 
## + Fold06.Rep4: alpha=0.65727, lambda=0.001635 
## - Fold06.Rep4: alpha=0.65727, lambda=0.001635 
## + Fold06.Rep4: alpha=0.23136, lambda=0.011723 
## - Fold06.Rep4: alpha=0.23136, lambda=0.011723 
## + Fold06.Rep4: alpha=0.08606, lambda=0.018838 
## - Fold06.Rep4: alpha=0.08606, lambda=0.018838 
## + Fold06.Rep4: alpha=0.27970, lambda=0.518103 
## - Fold06.Rep4: alpha=0.27970, lambda=0.518103 
## + Fold07.Rep4: alpha=0.62989, lambda=0.003636 
## - Fold07.Rep4: alpha=0.62989, lambda=0.003636 
## + Fold07.Rep4: alpha=0.29776, lambda=2.394057 
## - Fold07.Rep4: alpha=0.29776, lambda=2.394057 
## + Fold07.Rep4: alpha=0.28799, lambda=0.824474 
## - Fold07.Rep4: alpha=0.28799, lambda=0.824474 
## + Fold07.Rep4: alpha=0.94218, lambda=2.531489 
## - Fold07.Rep4: alpha=0.94218, lambda=2.531489 
## + Fold07.Rep4: alpha=0.35679, lambda=1.223740 
## - Fold07.Rep4: alpha=0.35679, lambda=1.223740 
## + Fold07.Rep4: alpha=0.63956, lambda=3.222801 
## - Fold07.Rep4: alpha=0.63956, lambda=3.222801 
## + Fold07.Rep4: alpha=0.65727, lambda=0.001635 
## - Fold07.Rep4: alpha=0.65727, lambda=0.001635 
## + Fold07.Rep4: alpha=0.23136, lambda=0.011723 
## - Fold07.Rep4: alpha=0.23136, lambda=0.011723 
## + Fold07.Rep4: alpha=0.08606, lambda=0.018838 
## - Fold07.Rep4: alpha=0.08606, lambda=0.018838 
## + Fold07.Rep4: alpha=0.27970, lambda=0.518103 
## - Fold07.Rep4: alpha=0.27970, lambda=0.518103 
## + Fold08.Rep4: alpha=0.62989, lambda=0.003636 
## - Fold08.Rep4: alpha=0.62989, lambda=0.003636 
## + Fold08.Rep4: alpha=0.29776, lambda=2.394057 
## - Fold08.Rep4: alpha=0.29776, lambda=2.394057 
## + Fold08.Rep4: alpha=0.28799, lambda=0.824474 
## - Fold08.Rep4: alpha=0.28799, lambda=0.824474 
## + Fold08.Rep4: alpha=0.94218, lambda=2.531489 
## - Fold08.Rep4: alpha=0.94218, lambda=2.531489 
## + Fold08.Rep4: alpha=0.35679, lambda=1.223740 
## - Fold08.Rep4: alpha=0.35679, lambda=1.223740 
## + Fold08.Rep4: alpha=0.63956, lambda=3.222801 
## - Fold08.Rep4: alpha=0.63956, lambda=3.222801 
## + Fold08.Rep4: alpha=0.65727, lambda=0.001635 
## - Fold08.Rep4: alpha=0.65727, lambda=0.001635 
## + Fold08.Rep4: alpha=0.23136, lambda=0.011723 
## - Fold08.Rep4: alpha=0.23136, lambda=0.011723 
## + Fold08.Rep4: alpha=0.08606, lambda=0.018838 
## - Fold08.Rep4: alpha=0.08606, lambda=0.018838 
## + Fold08.Rep4: alpha=0.27970, lambda=0.518103 
## - Fold08.Rep4: alpha=0.27970, lambda=0.518103 
## + Fold09.Rep4: alpha=0.62989, lambda=0.003636 
## - Fold09.Rep4: alpha=0.62989, lambda=0.003636 
## + Fold09.Rep4: alpha=0.29776, lambda=2.394057 
## - Fold09.Rep4: alpha=0.29776, lambda=2.394057 
## + Fold09.Rep4: alpha=0.28799, lambda=0.824474 
## - Fold09.Rep4: alpha=0.28799, lambda=0.824474 
## + Fold09.Rep4: alpha=0.94218, lambda=2.531489 
## - Fold09.Rep4: alpha=0.94218, lambda=2.531489 
## + Fold09.Rep4: alpha=0.35679, lambda=1.223740 
## - Fold09.Rep4: alpha=0.35679, lambda=1.223740 
## + Fold09.Rep4: alpha=0.63956, lambda=3.222801 
## - Fold09.Rep4: alpha=0.63956, lambda=3.222801 
## + Fold09.Rep4: alpha=0.65727, lambda=0.001635 
## - Fold09.Rep4: alpha=0.65727, lambda=0.001635 
## + Fold09.Rep4: alpha=0.23136, lambda=0.011723 
## - Fold09.Rep4: alpha=0.23136, lambda=0.011723 
## + Fold09.Rep4: alpha=0.08606, lambda=0.018838 
## - Fold09.Rep4: alpha=0.08606, lambda=0.018838 
## + Fold09.Rep4: alpha=0.27970, lambda=0.518103 
## - Fold09.Rep4: alpha=0.27970, lambda=0.518103 
## + Fold10.Rep4: alpha=0.62989, lambda=0.003636 
## - Fold10.Rep4: alpha=0.62989, lambda=0.003636 
## + Fold10.Rep4: alpha=0.29776, lambda=2.394057 
## - Fold10.Rep4: alpha=0.29776, lambda=2.394057 
## + Fold10.Rep4: alpha=0.28799, lambda=0.824474 
## - Fold10.Rep4: alpha=0.28799, lambda=0.824474 
## + Fold10.Rep4: alpha=0.94218, lambda=2.531489 
## - Fold10.Rep4: alpha=0.94218, lambda=2.531489 
## + Fold10.Rep4: alpha=0.35679, lambda=1.223740 
## - Fold10.Rep4: alpha=0.35679, lambda=1.223740 
## + Fold10.Rep4: alpha=0.63956, lambda=3.222801 
## - Fold10.Rep4: alpha=0.63956, lambda=3.222801 
## + Fold10.Rep4: alpha=0.65727, lambda=0.001635 
## - Fold10.Rep4: alpha=0.65727, lambda=0.001635 
## + Fold10.Rep4: alpha=0.23136, lambda=0.011723 
## - Fold10.Rep4: alpha=0.23136, lambda=0.011723 
## + Fold10.Rep4: alpha=0.08606, lambda=0.018838 
## - Fold10.Rep4: alpha=0.08606, lambda=0.018838 
## + Fold10.Rep4: alpha=0.27970, lambda=0.518103 
## - Fold10.Rep4: alpha=0.27970, lambda=0.518103 
## + Fold01.Rep5: alpha=0.62989, lambda=0.003636 
## - Fold01.Rep5: alpha=0.62989, lambda=0.003636 
## + Fold01.Rep5: alpha=0.29776, lambda=2.394057 
## - Fold01.Rep5: alpha=0.29776, lambda=2.394057 
## + Fold01.Rep5: alpha=0.28799, lambda=0.824474 
## - Fold01.Rep5: alpha=0.28799, lambda=0.824474 
## + Fold01.Rep5: alpha=0.94218, lambda=2.531489 
## - Fold01.Rep5: alpha=0.94218, lambda=2.531489 
## + Fold01.Rep5: alpha=0.35679, lambda=1.223740 
## - Fold01.Rep5: alpha=0.35679, lambda=1.223740 
## + Fold01.Rep5: alpha=0.63956, lambda=3.222801 
## - Fold01.Rep5: alpha=0.63956, lambda=3.222801 
## + Fold01.Rep5: alpha=0.65727, lambda=0.001635 
## - Fold01.Rep5: alpha=0.65727, lambda=0.001635 
## + Fold01.Rep5: alpha=0.23136, lambda=0.011723 
## - Fold01.Rep5: alpha=0.23136, lambda=0.011723 
## + Fold01.Rep5: alpha=0.08606, lambda=0.018838 
## - Fold01.Rep5: alpha=0.08606, lambda=0.018838 
## + Fold01.Rep5: alpha=0.27970, lambda=0.518103 
## - Fold01.Rep5: alpha=0.27970, lambda=0.518103 
## + Fold02.Rep5: alpha=0.62989, lambda=0.003636 
## - Fold02.Rep5: alpha=0.62989, lambda=0.003636 
## + Fold02.Rep5: alpha=0.29776, lambda=2.394057 
## - Fold02.Rep5: alpha=0.29776, lambda=2.394057 
## + Fold02.Rep5: alpha=0.28799, lambda=0.824474 
## - Fold02.Rep5: alpha=0.28799, lambda=0.824474 
## + Fold02.Rep5: alpha=0.94218, lambda=2.531489 
## - Fold02.Rep5: alpha=0.94218, lambda=2.531489 
## + Fold02.Rep5: alpha=0.35679, lambda=1.223740 
## - Fold02.Rep5: alpha=0.35679, lambda=1.223740 
## + Fold02.Rep5: alpha=0.63956, lambda=3.222801 
## - Fold02.Rep5: alpha=0.63956, lambda=3.222801 
## + Fold02.Rep5: alpha=0.65727, lambda=0.001635 
## - Fold02.Rep5: alpha=0.65727, lambda=0.001635 
## + Fold02.Rep5: alpha=0.23136, lambda=0.011723 
## - Fold02.Rep5: alpha=0.23136, lambda=0.011723 
## + Fold02.Rep5: alpha=0.08606, lambda=0.018838 
## - Fold02.Rep5: alpha=0.08606, lambda=0.018838 
## + Fold02.Rep5: alpha=0.27970, lambda=0.518103 
## - Fold02.Rep5: alpha=0.27970, lambda=0.518103 
## + Fold03.Rep5: alpha=0.62989, lambda=0.003636 
## - Fold03.Rep5: alpha=0.62989, lambda=0.003636 
## + Fold03.Rep5: alpha=0.29776, lambda=2.394057 
## - Fold03.Rep5: alpha=0.29776, lambda=2.394057 
## + Fold03.Rep5: alpha=0.28799, lambda=0.824474 
## - Fold03.Rep5: alpha=0.28799, lambda=0.824474 
## + Fold03.Rep5: alpha=0.94218, lambda=2.531489 
## - Fold03.Rep5: alpha=0.94218, lambda=2.531489 
## + Fold03.Rep5: alpha=0.35679, lambda=1.223740 
## - Fold03.Rep5: alpha=0.35679, lambda=1.223740 
## + Fold03.Rep5: alpha=0.63956, lambda=3.222801 
## - Fold03.Rep5: alpha=0.63956, lambda=3.222801 
## + Fold03.Rep5: alpha=0.65727, lambda=0.001635 
## - Fold03.Rep5: alpha=0.65727, lambda=0.001635 
## + Fold03.Rep5: alpha=0.23136, lambda=0.011723 
## - Fold03.Rep5: alpha=0.23136, lambda=0.011723 
## + Fold03.Rep5: alpha=0.08606, lambda=0.018838 
## - Fold03.Rep5: alpha=0.08606, lambda=0.018838 
## + Fold03.Rep5: alpha=0.27970, lambda=0.518103 
## - Fold03.Rep5: alpha=0.27970, lambda=0.518103 
## + Fold04.Rep5: alpha=0.62989, lambda=0.003636 
## - Fold04.Rep5: alpha=0.62989, lambda=0.003636 
## + Fold04.Rep5: alpha=0.29776, lambda=2.394057 
## - Fold04.Rep5: alpha=0.29776, lambda=2.394057 
## + Fold04.Rep5: alpha=0.28799, lambda=0.824474 
## - Fold04.Rep5: alpha=0.28799, lambda=0.824474 
## + Fold04.Rep5: alpha=0.94218, lambda=2.531489 
## - Fold04.Rep5: alpha=0.94218, lambda=2.531489 
## + Fold04.Rep5: alpha=0.35679, lambda=1.223740 
## - Fold04.Rep5: alpha=0.35679, lambda=1.223740 
## + Fold04.Rep5: alpha=0.63956, lambda=3.222801 
## - Fold04.Rep5: alpha=0.63956, lambda=3.222801 
## + Fold04.Rep5: alpha=0.65727, lambda=0.001635 
## - Fold04.Rep5: alpha=0.65727, lambda=0.001635 
## + Fold04.Rep5: alpha=0.23136, lambda=0.011723 
## - Fold04.Rep5: alpha=0.23136, lambda=0.011723 
## + Fold04.Rep5: alpha=0.08606, lambda=0.018838 
## - Fold04.Rep5: alpha=0.08606, lambda=0.018838 
## + Fold04.Rep5: alpha=0.27970, lambda=0.518103 
## - Fold04.Rep5: alpha=0.27970, lambda=0.518103 
## + Fold05.Rep5: alpha=0.62989, lambda=0.003636 
## - Fold05.Rep5: alpha=0.62989, lambda=0.003636 
## + Fold05.Rep5: alpha=0.29776, lambda=2.394057 
## - Fold05.Rep5: alpha=0.29776, lambda=2.394057 
## + Fold05.Rep5: alpha=0.28799, lambda=0.824474 
## - Fold05.Rep5: alpha=0.28799, lambda=0.824474 
## + Fold05.Rep5: alpha=0.94218, lambda=2.531489 
## - Fold05.Rep5: alpha=0.94218, lambda=2.531489 
## + Fold05.Rep5: alpha=0.35679, lambda=1.223740 
## - Fold05.Rep5: alpha=0.35679, lambda=1.223740 
## + Fold05.Rep5: alpha=0.63956, lambda=3.222801 
## - Fold05.Rep5: alpha=0.63956, lambda=3.222801 
## + Fold05.Rep5: alpha=0.65727, lambda=0.001635 
## - Fold05.Rep5: alpha=0.65727, lambda=0.001635 
## + Fold05.Rep5: alpha=0.23136, lambda=0.011723 
## - Fold05.Rep5: alpha=0.23136, lambda=0.011723 
## + Fold05.Rep5: alpha=0.08606, lambda=0.018838 
## - Fold05.Rep5: alpha=0.08606, lambda=0.018838 
## + Fold05.Rep5: alpha=0.27970, lambda=0.518103 
## - Fold05.Rep5: alpha=0.27970, lambda=0.518103 
## + Fold06.Rep5: alpha=0.62989, lambda=0.003636 
## - Fold06.Rep5: alpha=0.62989, lambda=0.003636 
## + Fold06.Rep5: alpha=0.29776, lambda=2.394057 
## - Fold06.Rep5: alpha=0.29776, lambda=2.394057 
## + Fold06.Rep5: alpha=0.28799, lambda=0.824474 
## - Fold06.Rep5: alpha=0.28799, lambda=0.824474 
## + Fold06.Rep5: alpha=0.94218, lambda=2.531489 
## - Fold06.Rep5: alpha=0.94218, lambda=2.531489 
## + Fold06.Rep5: alpha=0.35679, lambda=1.223740 
## - Fold06.Rep5: alpha=0.35679, lambda=1.223740 
## + Fold06.Rep5: alpha=0.63956, lambda=3.222801 
## - Fold06.Rep5: alpha=0.63956, lambda=3.222801 
## + Fold06.Rep5: alpha=0.65727, lambda=0.001635 
## - Fold06.Rep5: alpha=0.65727, lambda=0.001635 
## + Fold06.Rep5: alpha=0.23136, lambda=0.011723 
## - Fold06.Rep5: alpha=0.23136, lambda=0.011723 
## + Fold06.Rep5: alpha=0.08606, lambda=0.018838 
## - Fold06.Rep5: alpha=0.08606, lambda=0.018838 
## + Fold06.Rep5: alpha=0.27970, lambda=0.518103 
## - Fold06.Rep5: alpha=0.27970, lambda=0.518103 
## + Fold07.Rep5: alpha=0.62989, lambda=0.003636 
## - Fold07.Rep5: alpha=0.62989, lambda=0.003636 
## + Fold07.Rep5: alpha=0.29776, lambda=2.394057 
## - Fold07.Rep5: alpha=0.29776, lambda=2.394057 
## + Fold07.Rep5: alpha=0.28799, lambda=0.824474 
## - Fold07.Rep5: alpha=0.28799, lambda=0.824474 
## + Fold07.Rep5: alpha=0.94218, lambda=2.531489 
## - Fold07.Rep5: alpha=0.94218, lambda=2.531489 
## + Fold07.Rep5: alpha=0.35679, lambda=1.223740 
## - Fold07.Rep5: alpha=0.35679, lambda=1.223740 
## + Fold07.Rep5: alpha=0.63956, lambda=3.222801 
## - Fold07.Rep5: alpha=0.63956, lambda=3.222801 
## + Fold07.Rep5: alpha=0.65727, lambda=0.001635 
## - Fold07.Rep5: alpha=0.65727, lambda=0.001635 
## + Fold07.Rep5: alpha=0.23136, lambda=0.011723 
## - Fold07.Rep5: alpha=0.23136, lambda=0.011723 
## + Fold07.Rep5: alpha=0.08606, lambda=0.018838 
## - Fold07.Rep5: alpha=0.08606, lambda=0.018838 
## + Fold07.Rep5: alpha=0.27970, lambda=0.518103 
## - Fold07.Rep5: alpha=0.27970, lambda=0.518103 
## + Fold08.Rep5: alpha=0.62989, lambda=0.003636 
## - Fold08.Rep5: alpha=0.62989, lambda=0.003636 
## + Fold08.Rep5: alpha=0.29776, lambda=2.394057 
## - Fold08.Rep5: alpha=0.29776, lambda=2.394057 
## + Fold08.Rep5: alpha=0.28799, lambda=0.824474 
## - Fold08.Rep5: alpha=0.28799, lambda=0.824474 
## + Fold08.Rep5: alpha=0.94218, lambda=2.531489 
## - Fold08.Rep5: alpha=0.94218, lambda=2.531489 
## + Fold08.Rep5: alpha=0.35679, lambda=1.223740 
## - Fold08.Rep5: alpha=0.35679, lambda=1.223740 
## + Fold08.Rep5: alpha=0.63956, lambda=3.222801 
## - Fold08.Rep5: alpha=0.63956, lambda=3.222801 
## + Fold08.Rep5: alpha=0.65727, lambda=0.001635 
## - Fold08.Rep5: alpha=0.65727, lambda=0.001635 
## + Fold08.Rep5: alpha=0.23136, lambda=0.011723 
## - Fold08.Rep5: alpha=0.23136, lambda=0.011723 
## + Fold08.Rep5: alpha=0.08606, lambda=0.018838 
## - Fold08.Rep5: alpha=0.08606, lambda=0.018838 
## + Fold08.Rep5: alpha=0.27970, lambda=0.518103 
## - Fold08.Rep5: alpha=0.27970, lambda=0.518103 
## + Fold09.Rep5: alpha=0.62989, lambda=0.003636 
## - Fold09.Rep5: alpha=0.62989, lambda=0.003636 
## + Fold09.Rep5: alpha=0.29776, lambda=2.394057 
## - Fold09.Rep5: alpha=0.29776, lambda=2.394057 
## + Fold09.Rep5: alpha=0.28799, lambda=0.824474 
## - Fold09.Rep5: alpha=0.28799, lambda=0.824474 
## + Fold09.Rep5: alpha=0.94218, lambda=2.531489 
## - Fold09.Rep5: alpha=0.94218, lambda=2.531489 
## + Fold09.Rep5: alpha=0.35679, lambda=1.223740 
## - Fold09.Rep5: alpha=0.35679, lambda=1.223740 
## + Fold09.Rep5: alpha=0.63956, lambda=3.222801 
## - Fold09.Rep5: alpha=0.63956, lambda=3.222801 
## + Fold09.Rep5: alpha=0.65727, lambda=0.001635 
## - Fold09.Rep5: alpha=0.65727, lambda=0.001635 
## + Fold09.Rep5: alpha=0.23136, lambda=0.011723 
## - Fold09.Rep5: alpha=0.23136, lambda=0.011723 
## + Fold09.Rep5: alpha=0.08606, lambda=0.018838 
## - Fold09.Rep5: alpha=0.08606, lambda=0.018838 
## + Fold09.Rep5: alpha=0.27970, lambda=0.518103 
## - Fold09.Rep5: alpha=0.27970, lambda=0.518103 
## + Fold10.Rep5: alpha=0.62989, lambda=0.003636 
## - Fold10.Rep5: alpha=0.62989, lambda=0.003636 
## + Fold10.Rep5: alpha=0.29776, lambda=2.394057 
## - Fold10.Rep5: alpha=0.29776, lambda=2.394057 
## + Fold10.Rep5: alpha=0.28799, lambda=0.824474 
## - Fold10.Rep5: alpha=0.28799, lambda=0.824474 
## + Fold10.Rep5: alpha=0.94218, lambda=2.531489 
## - Fold10.Rep5: alpha=0.94218, lambda=2.531489 
## + Fold10.Rep5: alpha=0.35679, lambda=1.223740 
## - Fold10.Rep5: alpha=0.35679, lambda=1.223740 
## + Fold10.Rep5: alpha=0.63956, lambda=3.222801 
## - Fold10.Rep5: alpha=0.63956, lambda=3.222801 
## + Fold10.Rep5: alpha=0.65727, lambda=0.001635 
## - Fold10.Rep5: alpha=0.65727, lambda=0.001635 
## + Fold10.Rep5: alpha=0.23136, lambda=0.011723 
## - Fold10.Rep5: alpha=0.23136, lambda=0.011723 
## + Fold10.Rep5: alpha=0.08606, lambda=0.018838 
## - Fold10.Rep5: alpha=0.08606, lambda=0.018838 
## + Fold10.Rep5: alpha=0.27970, lambda=0.518103 
## - Fold10.Rep5: alpha=0.27970, lambda=0.518103

## Warning in nominalTrainWorkflow(x = x, y = y, wts = weights, info = trainInfo, :
## There were missing values in resampled performance measures.

## Aggregating results
## Selecting tuning parameters
## Fitting alpha = 0.0861, lambda = 0.0188 on full training set

# Best tuning parameter
elastic_reg$bestTune

##        alpha     lambda
## 1 0.08605824 0.01883794

Metrics for elasticnet

# Make predictions on training set
predictions_train <- predict(elastic_reg, x)
eval_results(y_train, predictions_train, train) 

##        RMSE   Rsquare
## 1 0.6338215 0.3638118

# Make predictions on test set
predictions_test <- predict(elastic_reg, x_test)
eval_results(y_test, predictions_test, test)

##        RMSE   Rsquare
## 1 0.6757623 0.3456848

colnames(df)

##  [1] "fixed.acidity"        "volatile.acidity"     "citric.acid"         
##  [4] "residual.sugar"       "chlorides"            "free.sulfur.dioxide" 
##  [7] "total.sulfur.dioxide" "density"              "pH"                  
## [10] "sulphates"            "alcohol"              "quality"

ggplot(data = df, mapping = aes(x = total.sulfur.dioxide, y = quality)) + 
  geom_point()

ggplot(data = df, mapping = aes(x = residual.sugar, y = quality)) + 
  geom_point()

There is really no relationship. This is probably an exercise in clusterting or machine learning, but linear regression is insufficient.